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RD Sharma solutions for Class 11 Mathematics chapter 19 - Arithmetic Progression

Mathematics Class 11

RD Sharma Mathematics Class 11 Chapter 19: Arithmetic Progression

Ex. 19.10Ex. 19.20Ex. 19.40Ex. 19.50Ex. 19.60Ex. 5.60Ex. 19.70Others

Chapter 19: Arithmetic Progression Exercise 19.10 solutions [Page 4]

Ex. 19.10 | Q 1 | Page 4

If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.

Ex. 19.10 | Q 2 | Page 4

A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.

Ex. 19.10 | Q 3 | Page 4

Let < an > be a sequence defined by a1 = 3 and, an = 3an − 1 + 2, for all n > 1
Find the first four terms of the sequence.

Ex. 19.10 | Q 4.1 | Page 4

Let < an > be a sequence. Write the first five term in the following:

a1 = 1, an = an − 1 + 2, n ≥ 2

Ex. 19.10 | Q 4.2 | Page 4

Let < an > be a sequence. Write the first five term in the following:

a1 = 1 = a2, an = an − 1 + an − 2, n > 2

Ex. 19.10 | Q 4.3 | Page 4

Let < an > be a sequence. Write the first five term in the following:

a1 = a2 = 2, an = a− 1 − 1, n > 2

Ex. 19.10 | Q 5 | Page 4

The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2

Find (""^an +1)/(""^an") for n = 1, 2, 3, 4, 5.

Ex. 19.10 | Q 6.1 | Page 4

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

3, −1, −5, −9 ...

Ex. 19.10 | Q 6.2 | Page 4

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

−1, 1/4, 3/2, 11/4, ...

Ex. 19.10 | Q 6.3 | Page 4

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

$\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .$

Ex. 19.10 | Q 6.4 | Page 4

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

9, 7, 5, 3, ...

Ex. 19.10 | Q 7 | Page 4

The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.

Ex. 19.10 | Q 8 | Page 4

The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.

Chapter 19: Arithmetic Progression Exercise 19.20 solutions [Pages 11 - 12]

Ex. 19.20 | Q 1.1 | Page 11

Find:

10th term of the A.P. 1, 4, 7, 10, ...

Ex. 19.20 | Q 1.2 | Page 11

Find:

18th term of the A.P.

$\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},$

Ex. 19.20 | Q 1.3 | Page 11

Find:

nth term of the A.P. 13, 8, 3, −2, ...

Ex. 19.20 | Q 2 | Page 12

If the sequence < an > is an A.P., show that am +n +am − n = 2am.

Ex. 19.20 | Q 3.1 | Page 12

Which term of the A.P. 3, 8, 13, ... is 248?

Ex. 19.20 | Q 3.2 | Page 12

Which term of the A.P. 84, 80, 76, ... is 0?

Ex. 19.20 | Q 3.3 | Page 12

Which term of the A.P. 4, 9, 14, ... is 254?

Ex. 19.20 | Q 4.1 | Page 12

Is 68 a term of the A.P. 7, 10, 13, ...?

Ex. 19.20 | Q 4.2 | Page 12

Is 302 a term of the A.P. 3, 8, 13, ...?

Ex. 19.20 | Q 5.1 | Page 12

Which term of the sequence 24, $23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}$....... is the first negative term?

Ex. 19.20 | Q 5.2 | Page 12

Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?

Ex. 19.20 | Q 5.3 | Page 12

Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?

Ex. 19.20 | Q 6.1 | Page 12

How many terms are there in the A.P. 7, 10, 13, ... 43 ?

Ex. 19.20 | Q 6.2 | Page 12

How many terms are there in the A.P.$- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?$

Ex. 19.20 | Q 7 | Page 12

The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.

Ex. 19.20 | Q 8 | Page 12

The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.

Ex. 19.20 | Q 9 | Page 12

If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.

Ex. 19.20 | Q 10 | Page 12

If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

Ex. 19.20 | Q 11 | Page 12

The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.

Ex. 19.20 | Q 12 | Page 12

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Ex. 19.20 | Q 13 | Page 12

If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.

Ex. 19.20 | Q 14 | Page 12

If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.

Ex. 19.20 | Q 15.1 | Page 12

Find the 12th term from the following arithmetic progression:

3, 5, 7, 9, ... 201

Ex. 19.20 | Q 15.2 | Page 12

Find the 12th term from the following arithmetic progression:

3, 8, 13, ..., 253

Ex. 19.20 | Q 15.3 | Page 12

Find the 12th term from the following arithmetic progression:

1, 4, 7, 10, ..., 88

Ex. 19.20 | Q 16 | Page 12

The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.

Ex. 19.20 | Q 17 | Page 12

Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.

Ex. 19.20 | Q 18 | Page 12

How many numbers of two digit are divisible by 3?

Ex. 19.20 | Q 19 | Page 12

An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.

Ex. 19.20 | Q 20 | Page 12

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Ex. 19.20 | Q 21 | Page 12

How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?

Ex. 19.20 | Q 22 | Page 12

The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.

Ex. 19.20 | Q 23 | Page 12

If < an > is an A.P. such that $\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}$.

Ex. 19.20 | Q 24 | Page 12

$\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }$

$\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]$

Chapter 19: Arithmetic Progression Exercise 19.20 solutions [Page 15]

Ex. 19.20 | Q 1 | Page 15

The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.

Ex. 19.20 | Q 2 | Page 15

Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.

Ex. 19.20 | Q 3 | Page 15

Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

Ex. 19.20 | Q 4 | Page 15

The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.

Ex. 19.20 | Q 5 | Page 15

If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.

Ex. 19.20 | Q 6 | Page 15

The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.

Chapter 19: Arithmetic Progression Exercise 19.40 solutions [Pages 30 - 31]

Ex. 19.40 | Q 1.1 | Page 30

Find the sum of the following arithmetic progression :

50, 46, 42, ... to 10 terms

Ex. 19.40 | Q 1.2 | Page 30

Find the sum of the following arithmetic progression :

1, 3, 5, 7, ... to 12 terms

Ex. 19.40 | Q 1.3 | Page 30

Find the sum of the following arithmetic progression :

3, 9/2, 6, 15/2, ... to 25 terms

Ex. 19.40 | Q 1.4 | Page 30

Find the sum of the following arithmetic progression :

41, 36, 31, ... to 12 terms

Ex. 19.40 | Q 1.5 | Page 30

Find the sum of the following arithmetic progression :

a + b, a − b, a − 3b, ... to 22 terms

Ex. 19.40 | Q 1.6 | Page 30

Find the sum of the following arithmetic progression :

(x − y)2, (x2 + y2), (x + y)2, ... to n terms

Ex. 19.40 | Q 1.7 | Page 30

Find the sum of the following arithmetic progression :

$\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}$, ... to n terms.

Ex. 19.40 | Q 2.1 | Page 30

Find the sum of the following serie:

2 + 5 + 8 + ... + 182

Ex. 19.40 | Q 2.2 | Page 30

Find the sum of the following serie:

101 + 99 + 97 + ... + 47

Ex. 19.40 | Q 2.3 | Page 30

Find the sum of the following serie:

(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]

Ex. 19.40 | Q 3 | Page 30

Find the sum of first n natural numbers.

Ex. 19.40 | Q 4 | Page 30

Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.

Ex. 19.40 | Q 5 | Page 30

Find the sum of first n odd natural numbers.

Ex. 19.40 | Q 6 | Page 30

Find the sum of all odd numbers between 100 and 200.

Ex. 19.40 | Q 7 | Page 30

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Ex. 19.40 | Q 8 | Page 30

Find the sum of all integers between 84 and 719, which are multiples of 5.

Ex. 19.40 | Q 9 | Page 31

Find the sum of all integers between 50 and 500 which are divisible by 7.

Ex. 19.40 | Q 10 | Page 31

Find the sum of all even integers between 101 and 999.

Ex. 19.40 | Q 11 | Page 31

Find the sum of all integers between 100 and 550, which are divisible by 9.

Ex. 19.40 | Q 12 | Page 31

Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.

Ex. 19.40 | Q 13 | Page 31

Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.

Ex. 19.40 | Q 14.1 | Page 31

Solve:

25 + 22 + 19 + 16 + ... + x = 115

Ex. 19.40 | Q 14.2 | Page 31

Solve:

1 + 4 + 7 + 10 + ... + x = 590.

Ex. 19.40 | Q 15 | Page 31

Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.

Ex. 19.40 | Q 16 | Page 31

How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?

Ex. 19.40 | Q 17 | Page 31

The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.

Ex. 19.40 | Q 18 | Page 31

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

Ex. 19.40 | Q 19 | Page 31

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

Ex. 19.40 | Q 20 | Page 31

The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by $10 \frac{1}{2}$ ,find the number of terms and the series.

Ex. 19.40 | Q 21 | Page 31

If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.

Ex. 19.40 | Q 22 | Page 31

If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

Ex. 19.40 | Q 23 | Page 31

If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?

Ex. 19.40 | Q 24 | Page 31

Find the sum of n terms of the A.P. whose kth terms is 5k + 1.

Ex. 19.40 | Q 25 | Page 31

Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

Ex. 19.40 | Q 26 | Page 31

If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.

Ex. 19.40 | Q 27 | Page 31

Find the sum of odd integers from 1 to 2001.

Ex. 19.40 | Q 28 | Page 31

How many terms of the A.P. −6, $- \frac{11}{2}$, −5, ... are needed to give the sum −25?

Ex. 19.40 | Q 29 | Page 31

In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.

Ex. 19.40 | Q 30 | Page 31

If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, the prove that:
S1 : S2 = (2n + 1) : (n + 1)

Ex. 19.40 | Q 31 | Page 31

Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.

Ex. 19.40 | Q 32 | Page 31

If the sum of n terms of an A.P. is nP + $\frac{1}{2}$ n (n − 1) Q, where P and Q are constants, find the common difference.

Ex. 19.40 | Q 33 | Page 31

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.

Ex. 19.40 | Q 34 | Page 31

The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.

Chapter 19: Arithmetic Progression Exercise 19.50 solutions [Page 42]

Ex. 19.50 | Q 1.1 | Page 42

If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P., prove that:

$\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}$ are in A.P.

Ex. 19.50 | Q 1.2 | Page 42

If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P., prove that:

a (b +c), b (c + a), c (a +b) are in A.P.

Ex. 19.50 | Q 2 | Page 42

If a2, b2, c2 are in A.P., prove that $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are in A.P.

Ex. 19.50 | Q 3.1 | Page 42

If a, b, c is in A.P., then show that:

a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.

Ex. 19.50 | Q 3.2 | Page 42

If a, b, c is in A.P., then show that:

b + c − a, c + a − b, a + b − c are in A.P.

Ex. 19.50 | Q 3.3 | Page 42

If a, b, c is in A.P., then show that:

bc − a2, ca − b2, ab − c2 are in A.P.

Ex. 19.50 | Q 4.1 | Page 42

If $\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}$ are in A.P., prove that:

$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.

Ex. 19.50 | Q 4.2 | Page 42

If $\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}$ are in A.P., prove that:

bc, ca, ab are in A.P.

Ex. 19.50 | Q 5.1 | Page 42

If a, b, c is in A.P., prove that:

(a − c)2 = 4 (a − b) (b − c)

Ex. 19.50 | Q 5.2 | Page 42

If a, b, c is in A.P., prove that:

a2 + c2 + 4ac = 2 (ab + bc + ca)

Ex. 19.50 | Q 5.3 | Page 42

If a, b, c is in A.P., prove that:

a3 + c3 + 6abc = 8b3.

Ex. 19.50 | Q 6 | Page 42

If $a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)$ are in A.P., prove that abc are in A.P.

Ex. 19.50 | Q 7 | Page 42

Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.

Chapter 19: Arithmetic Progression Exercise 19.60, 5.60 solutions [Pages 46 - 54]

Ex. 19.60 | Q 1.1 | Page 46

Find the A.M. between:

7 and 13

Ex. 19.60 | Q 1.2 | Page 46

Find the A.M. between:

12 and −8

Ex. 19.60 | Q 1.3 | Page 46

Find the A.M. between:

(x − y) and (x + y).

Ex. 19.60 | Q 2 | Page 46

Insert 4 A.M.s between 4 and 19.

Ex. 19.60 | Q 3 | Page 46

Insert 7 A.M.s between 2 and 17.

Ex. 19.60 | Q 4 | Page 46

Insert six A.M.s between 15 and −13.

Ex. 19.60 | Q 5 | Page 46

There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.

Ex. 19.60 | Q 6 | Page 46

Insert A.M.s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.s.

Ex. 19.60 | Q 7 | Page 46

If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

Ex. 19.60 | Q 8 | Page 47

If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.

Ex. 19.60 | Q 9 | Page 47

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Ex. 5.60 | Q 66 | Page 54

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

Chapter 19: Arithmetic Progression Exercise 19.70 solutions [Pages 49 - 50]

Ex. 19.70 | Q 1 | Page 49

A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?

Ex. 19.70 | Q 2 | Page 49

A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.

Ex. 19.70 | Q 3 | Page 49

A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.

Ex. 19.70 | Q 4 | Page 49

A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.

Ex. 19.70 | Q 5 | Page 49

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

Ex. 19.70 | Q 6 | Page 49

A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

Ex. 19.70 | Q 7 | Page 49

A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?

Ex. 19.70 | Q 8 | Page 49

A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?

Ex. 19.70 | Q 9 | Page 49

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.

Ex. 19.70 | Q 10 | Page 49

The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.

Ex. 19.70 | Q 11 | Page 49

A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?

Ex. 19.70 | Q 12 | Page 49

A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?

Ex. 19.70 | Q 13 | Page 50

We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.

Ex. 19.70 | Q 14 | Page 50

In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

Ex. 19.70 | Q 15 | Page 50

A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?

Ex. 19.70 | Q 16 | Page 50

A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?

Ex. 19.70 | Q 17 | Page 50

In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?

Chapter 19: Arithmetic Progression solutions [Pages 50 - 51]

Q 1 | Page 50

Write the common difference of an A.P. whose nth term is xn + y.

Q 2 | Page 50

Write the common difference of an A.P. the sum of whose first n terms is

$\frac{p}{2} n^2 + Qn$.
Q 3 | Page 50

If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.

Q 4 | Page 50

If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.

Q 5 | Page 50

If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.

Q 6 | Page 50

Write the sum of first n odd natural numbers.

Q 7 | Page 50

Write the sum of first n even natural numbers.

Q 8 | Page 50

Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.

Q 9 | Page 51

If $\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}$ 7, then find the value of n.

Q 10 | Page 51

If m th term of an A.P. is n and nth term is m, then write its pth term.

Q 11 | Page 51

If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.

Chapter 19: Arithmetic Progression solutions [Pages 51 - 53]

Q 1 | Page 51

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

• 87

• 88

• 89

• 90

Q 2 | Page 51

If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

• 0

•  p − q

• p + q

•  − (p + q)

Q 3 | Page 51

If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term is

• 2

• 3

• 1

• 4

Q 4 | Page 51

Sum of all two digit numbers which when divided by 4 yield unity as remainder is

• 1200

•  1210

• 1250

• none of these.

Q 5 | Page 51

In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is

• 6

• 8

• 4

•  none of these.

Q 6 | Page 51

If Sn denotes the sum of first n terms of an A.P. < an > such that

$\frac{S_m}{S_n} = \frac{m^2}{n^2}, \text { then }\frac{a_m}{a_n} =$
• $\frac{2 m + 1}{2 n + 1}$

• $\frac{2 m - 1}{2 n - 1}$

• $\frac{m - 1}{n - 1}$

• $\frac{m + 1}{n + 1}$

Q 7 | Page 51

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be

• 5

• 6

• 7

• 8

Q 8 | Page 51

If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?

•  26th

•  27th

•  28th

• none of these.

Q 9 | Page 51

If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is

• 4n − 3

• 3 n − 4

•  4 n + 3

• 3 n + 4

Q 10 | Page 51

If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is

•  sec a1 − sec an

• cosec a1 − cosec an

• cot a1 − cot an

• tan a1 − tan an

Q 11 | Page 52

In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is

•  1/5

•  2/3

• 3/4

• none of these

Q 12 | Page 52

If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is

•  sec a1 − sec an

• cosec a1 − cosec an

• cot a1 − cot an

• tan an − tan a1

Q 13 | Page 52

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

•  5, 10, 15, 20

• 4, 10, 16, 22

• 3, 7, 11, 15

•  none of these

Q 14 | Page 52

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is

• 10

• 12

• 13

• 14

Q 15 | Page 52

Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =

• 1

• 2

• 3

• none of these

Q 16 | Page 52

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by $\frac{l^2 - a^2}{k - (l + a)}$ ,  then k =

• S

• 2S

• 3S

• none of these

Q 17 | Page 52

If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =

• $\frac{1}{n}$

• $\frac{n - 1}{n}$

• $\frac{n + 1}{2n}$

• $\frac{n + 1}{n}$

Q 18 | Page 52

If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is

• $\frac{ab}{2 (b - a)}$

• $\frac{ab}{b - a}$

• $\frac{3 ab}{2 (b - a)}$

• none of these

Q 19 | Page 52

If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then $\frac{S_1}{S_2}$ =

• $\frac{2n}{n + 1}$

• $\frac{n}{n + 1}$

• $\frac{n + 1}{2n}$

• $\frac{n + 1}{n}$

Q 20 | Page 52

If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to

• $\frac{1}{2} p^3$

•  mn p

• P3

• (m + n) p2

Q 21 | Page 52

Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is

• $-$p

• p

• q

• p-q

Q 22 | Page 52

Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is

• 191

• 193

• 211

• none of these

Q 23 | Page 52

Mark the correct alternative in the following question:

$\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }$

• q^3/2

• mnq

• q^3

• (m^2+n^2)q

Q 24 | Page 53

Mark the correct alternative in the following question:

Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to

• 4

• 6

• 8

• 10

Chapter 19: Arithmetic Progression

Ex. 19.10Ex. 19.20Ex. 19.40Ex. 19.50Ex. 19.60Ex. 5.60Ex. 19.70Others

RD Sharma Mathematics Class 11 RD Sharma solutions for Class 11 Mathematics chapter 19 - Arithmetic Progression

RD Sharma solutions for Class 11 Maths chapter 19 (Arithmetic Progression) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 19 Arithmetic Progression are Sum to N Terms of Special Series, Introduction of Sequence and Series, Concept of Sequences, Concept of Series, Arithmetic Progression (A.P.), Geometric Progression (G. P.), Relationship Between A.M. and G.M..

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