R.D. Sharma solutions for Mathematics [English] Class 10 chapter 5 - Arithmetic Progressions [Latest edition]

Write the first three terms in the sequences defined by the following:

a_n = 3n + 2

Write the first five terms of the following sequences whose nth term is:

a_n = 3ⁿ

Write the first five terms of the following sequences whose nth term is:

a_n = (–1)ⁿ . 2ⁿ

Write the first five terms of the following sequences whose nth term is:

`a_n = (n(n - 2))/2`

Write the first five terms of the following sequences whose nth term is:

a_n = n² – n + 1

Write the first five terms of the following sequence whose nth term is:

a_n = 2n² – 3n + 1

Find the indicated terms in the following sequences whose nth term is:

a_n = 5n – 4; a₁₂ and a₁₅

Find the indicated terms in the following sequences whose nth term is:

`a_n = (3n - 2)/(4n + 5)`; `a_7 and a_8`

Find the indicated terms in the following sequences whose nth term is:

a_n = n (n – 1) (n – 2); a₅ and a₈

Find the indicated terms in the following sequences whose nth term is:

`a_n = (-1)^2 n; a_3, a_5, a_8`

Find the next five terms of the following sequences given by:

a₁ = 1, a_n = a_{n – 1} + 2, n ≥ 2

Find the next five terms of the following sequences given by:

a₁ = a₂ = 2, a_n = a_n−1 – 3, n > 2

Write the sequence with nth term:

a_n = 3 + 4n

Write the sequence with nth term:

a_n = 5 + 2n

Write the sequence with nth term:

a_n = 6 – n

Write the sequence with nth term: 

a_n = 9 – 5n

Show that the sequence defined by a_n = 5n – 7 is an A.P., find its common difference.

Show that the sequence defined by a_n = 3n² – 5 is not an A.P.

The general term of a sequence is given by a_n = – 4n + 15. Is the sequence an A.P.? If so, find its 15^th term and the common difference.

Justify whether it is true to say that the sequence having following n^th term is an A.P.

a_n = 2n – 1

Justify whether it is true to say that the sequence having following n^th term is an A.P.

a_n = 3n² + 5

Justify whether it is true to say that the sequence having following n^th term is an A.P.

a_n = 1 + n + n²

The n^th term of an A.P. cannot be n² + 1. Justify your answer.

For the following arithmetic progressions write the first term a and the common difference d:

 –5, –1, 3, 7, ...

For the following arithmetic progressions write the first term a and the common difference d:

`1/5, 3/5, 5/5, 7/5`

For the following arithmetic progressions write the first term a and the common difference d:

0.3, 0.55, 0.80, 1.05, ...

For the following arithmetic progressions write the first term a and the common difference d:

–1.1, –3.1, –5.1, –7.1, ...

Write the arithmetic progression when first term a and common difference d is as follows:

a = 4, d = –3

Write the arithmetic progression when first term a and common difference d is as follows:

`a = -1, d = 1/2`

Write the arithmetic progressions when first term a and common difference d is as follows:

a = –1.5, d = –0.5

In the following situation, the sequence of numbers formed will form an A.P.?

The cost of digging a well for the first metre is Rs 150 and rises by Rs 20 for each succeeding metre.

In the following situation, involved make an arithmetic progression? and why?

The amount of air present in a cylinder when a vacuum pump removes `1/4` of the air remaining in the cylinder at a time.

In the following situation, the sequence of numbers formed will form an A.P.?

Divya deposited ₹ 1000 at compound interest at the rate of 10% per annum. The amount at the end of first year, second year, third year, ..., and so on.

Find the common difference and write the next four terms of the following arithmetic progressions:

1, –2, –5, –8,...

Find the common difference and write the next four terms of the following arithmetic progressions:

0, –3, –6, –9,...

Find the common difference and write the next four terms of the following arithmetic progressions:

`-1, 1/4, 3/2,...`

Find the common difference and write the next four terms of the following arithmetic progressions:

`-1, -5/6, -2/3,...`

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

3, 3, 3, 3,...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

p, p + 90, p + 180 p + 270,... where p = (999)⁹⁹⁹

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

1.0, 1.7, 2.4, 3.1,...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

–225, –425, –625, –825,...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

1², 3², 5², 7²,...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2),....

Prove that no matter what the real numbers a and b are, the sequence with the nth term a + nb is always an A.P. What is the common difference?

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

Find the 18^th term of the A.P. `sqrt2, 3sqrt2, 5sqrt2,...`

Find the n^th term of the A.P. 13, 8, 3, –2,...

Find the 10^th term of the A.P. –40, –15, 10, 35,...

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

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

Which term of the A.P. : 65, 61, 57, 53, .............. is the first negative term?

Which term of the A.P. –7, –12, –17, –22,... will be –82? Is –100 any term of the A.P.?

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

Is –150 a term of the A.P. 11, 8, 5, 2,...?

How many terms are there in the A.P.?

7, 10, 13, ... 43

How many terms are there in the A.P.?

`-1, -5/6, -2/3, -1/2,...,10/3`

Find the number of terms in the following A.P.: 7, 13, 19, ..., 205.

Find the number of terms in the following A.P.

`18, 15 1/2, 13, ..., – 47`

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.

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

Find the 12^th term from the end of the following arithmetic progressions:

3, 8, 13, ..., 253

Find the 12^th term from the end of the following arithmetic progressions:

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

If the seventh term of an A.P. is \[\frac{1}{9}\] and its ninth term is \[\frac{1}{7}\], find its (63)^rd term. 

Which term of the A.P. 3, 15, 27, 39, ... will be 120 more than its 21^st term?

For what value of n, the n^th terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?

Find the arithmetic progression whose third term is 16 and the seventh term exceeds its fifth term by 12.

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

The sum of the 5^th and 7^th terms of an A.P. is 52 and the 10^th term is 46. Find the A.P.

If sum of 3^rd and 8^th terms of an A.P. is 7 and sum of 7^th and 14^th terms is –3 then find the 10^th term.

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

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

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

The 26^th, 11^th and the last term of an AP are 0, 3 and `- 1/5`, respectively. Find the common difference and the number of terms.

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

Write the expression a_n – a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which 11^th term is 5 and 13^th term is 79.

Write the expression a_n – a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which a₁₀ – a₅ = 200.

Write the expression a_n – a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which 20^th term is 10 more than the 18^th term.

The eighth term of an AP is half its second term and the eleventh term exceeds one third of its fourth term by 1. Find the 15^th term.

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

How many multiples of 4 lie between 10 and 250?

The 17^th term of an A.P. is 5 more than twice its 8^th term. If the 11^th term of the A.P. is 43, find the n^th term.

The 19^th term of an A.P. is equal to three times its sixth term. If its 9^th term is 19, find the A.P.

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Find the sum of two middle terms of the A.P. :

`-4/3, -1, (-2)/3, -1/3, ..., 4 1/3`

A sum of ₹ 2,000 is invested at 7% per annum simple interest. Calculate the interests at the end of 1^st, 2^nd and 3^rd year. Do these interests form an A.P.? If so, find the interest at the end of the 27^th year.

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.

If an A.P. consists of n terms with first term a and n^th term `l` show that the sum of the m^th term from the beginning and the m^th term from the end is (a + `l`).

How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?

For the AP: –3, –7, –11, ..., can we find directly a₃₀ – a₂₀ without actually finding a₃₀ and a₂₀? Give reasons for your answer.

Two APs have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10^th terms is the same as the difference between their 21^st terms, which is the same as the difference between any two corresponding terms. Why?

Find the value of x for which (8x + 4), (6x – 2) and (2x + 7) are in A.P.

If x + 1, 3x and 4x + 2 are in A.P., find the value of x.

Show that (a – b)², (a² + b²) and (a + b)² are in A.P.

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

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.

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

In a A.P., the sum of the three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.

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

Divide 56 in four parts in A.P. such that the ratio of the product of their extremes to the product of their means is 5 : 6.

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.

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

Split 207 into three parts such that these are in A.P. and the product of the two smaller parts is 4623.

The angles of a triangle are in AP. The greatest angle is twice the least. Find all the angles of the triangle.

Find the sum of the following arithmetic progressions:

–26, –24, –22,... to 36 terms

Find the sum of the following arithmetic progressions:

a + b, a – b, a – 3b, ... to 22 terms

Find the sum of the following arithmetic progressions:

`(x - y)^2, (x^2 + y^2), (x + y)^2` ...., to n terms

Find the sum of the following arithmetic progressions:

`(x - y)/(x + y), (3x - 2y)/(x + y), (5x - 3y)/(x + y),...` to n terms

Which term of the A.P. –2, –7, –12,..... will be –77? Find the sum of this A.P. upto the term –77.

Find the sum of last ten terms of the A.P.: 8, 10, 12, 14, .., 126.

Find the sum of the first 15 terms of the following sequences having n^th term as a_n = 3 + 4n.

Find the sum of the first 15 terms of the following sequences having n^th term as y_n = 9 – 5n.

Find the sum of first 20 terms of an A.P. whose n^th term is given as a_n = 5 – 2n.

If the sum of a certain number of terms starting from first term of an A.P. is 25, 22, 19, ..., is 116. Find the last term.

How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?

How many terms of the A.P. 27, 24, 21, ..., should be taken so that their sum is zero?

How many terms of the A.P. 45, 39, 33... must be taken so that their sum is 180? Explain the double answer.

Find the sum of first 15 multiples of 8.

Find the sum of the first 40 positive integers divisible by 3.

Find the sum of the first 40 positive integers divisible by 5.

Find the sum of first 40 positive integers divisible by 6.

Find the sum of all 3 – digit natural numbers which are divisible by 13.

Find the sum of all 3-digit natural numbers, which are multiples of 11.

Find the sum of all 2-digit natural numbers divisible by 4.

Find the sum of first 8 multiples of 3.

Find the sum given below:

34 + 32 + 30 + ... + 10

Find the sum:

25 + 28 + 31 + ... + 100

Find the sum:  

\[18 + 15\frac{1}{2} + 13 + ... + \left( - 49\frac{1}{2} \right)\]

The first and last terms of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there, and what is their sum?

Find the sum of n terms of the series \[\left( 4 - \frac{1}{n} \right) + \left( 4 - \frac{2}{n} \right) + \left( 4 - \frac{3}{n} \right) + ....\]

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

If the sum of first n terms of an A.P. is \[\frac{1}{2}\] (3n² + 7n), then find its n^th term. Hence write its 20^th term.

The sums of first n terms of three A.P.s are S₁, S₂ and S₃. The first term of each is 5 and their common differences are 2, 4 and 6 respectively. Prove that S₁ + S₃ = 2S₂.

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

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

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.

If 12^th 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?

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

If the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

​The first and the last terms of an A.P. are 7 and 49 respectively. If sum of all its terms is 420, find its common difference. 

The sum of first 9 terms of an A.P. is 162. The ratio of its 6^th term to its 13^th term is 1 : 2. Find the first and 15^th term of the A.P.

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28^th term of this A.P.

The nth term of an A.P. is given by (–4n + 15). Find the sum of first 20 terms of this A.P.

Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25th term.

The first and last terms of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there, and what is their sum?

The first term of an AP is –5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms and the common difference.

If the sum of the first n terms of an AP is 4n – n², what is the first term (that is, S₁)? What is the sum of the first two terms? What is the second term? Similarly, find the 3^rd, the 10th, and the n^th terms.

If the sum of first n terms of an AP is n², then find its 10th term.

If a_n = 3 – 4n, show that a₁, a₂, a₃,... form an AP. Also find S₂₀.

In an A.P., if S_n = n(4n + 1), find the A.P.

Find the sum of first seven numbers which are multiples of 2 as well as of 9.

Find the sum of the odd numbers between 0 and 50.

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

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

If sum of first 6 terms of an AP is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.

Find the sum of first 17 terms of an AP whose 4^th and 9^th terms are –15 and –30 respectively.

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

An A.P. consists of 'n' terms whose n^th term is 4 and the common difference is 2. If the sum of 'n' terms of A.P. is –14, then find 'n'. Also, find the sum of the first 20 terms.

A thief, after committing a theft, runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief?

Reshma wanted to save at least Rs 6,500 for sending her daughter to school next year (after 12 month.) She saved Rs 450 in the first month and raised her savings by Rs 20 every next month. How much will she be able to save in next 12 months? Will she be able to send her daughter to the school next year?

Ramkali would need ₹1800 for admission fee and books etc., for her daughter to start going to school from next year. She saved ₹ 50 in the first month of this year and increased her monthly saving by ₹ 20. After a year, how much money will she save? Will she be able to fulfil her dream of sending her daughter to school?

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?

A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years, and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.

Solve the equation:

– 4 + (–1) + 2 + 5 + ... + x = 437

The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is – 30 and the common difference is 8. Find n.

The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?

The ratio of the 11^th term to the 18^th term of an AP is 2 : 3. Find the ratio of the 5^th term to the 21^st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

The ratio of the 11^th term to 17^th term of an A.P. is 3 : 4. Find the ratio of 5^th term to 21^st term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

Find the common difference of the Arithmetic progression

`1/a, (3 - a)/(3a), (3 - 2a)/(3a),"......"(a ≠ 0)`

In an A.P., if the common difference d = – 4, and the seventh term a₇ is 4, then find the first term.

Write the nth term of the \[A.P. \frac{1}{m}, \frac{1 + m}{m}, \frac{1 + 2m}{m}, . . . .\]

For what value of k will the consecutive terms 2k + 1, 3k + 3 and 5k – 1 form an A.P.?

Find the 9^th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185

If `4/5`, a, 2 are three consecutive terms of an A.P., then find the value of a.

Write the expression for the common difference of an A.P. whose first term is a and nth term is b.

Find the sum of the first 10 multiples of 3.

How many two digits numbers are divisible by 3?

In an A.P. if the sum of third and seventh term is zero. Find its 5^th term.

Determine the A.P. whose third term is 5 and the seventh term is 9.

The sequence `(2a - 6b)/(3b), (2a - 3b)/(3b), (2a)/(3b), (2a + 3b)/(3b), (2a + 6b)/(3b)...` is an A.P. with common difference ______.

If the sequence `(a - 3b)/b, (3a - 3b)/b, (5a - 3b)/b, (7a - 3b)/b,...` is an A.P. with common difference 3, then `a/b` = ______.

The 11th term of the A.P. `-5, -5/2, 0, 5/2,` ...... is ______.

The 21^st term of the A.P. whose first two terms are –3 and 4 is ______.

The famous Mathematician associated with finding the sum of the first 100 natural numbers was ______.

If n – 2, 4n – 1 and 5n + 2 are in A.P., then n = ______.

The sum of first 50 odd natural numbers is ______.

The sum of first n odd natural numbers is ______.

If the common difference of an A.P. is 5, then a₁₈ – a₁₃ = ______.

If a₁, a₂, a₃,......, a_n,...... is an A.P. such that a₁₈ – a₁₄ = 32, then its common difference is ______.

If 5, a₂, a₃,...., a₂₀, 145 is an A.P., then a₂ + a₂₀ = ______.

The value of the middle term of the A.P. –11, –7, –3, ...., 49, 53 is ______.

If 9^th term of an A.P. is zero, then its 29^th and 19^th terms are in the ratio ______.

If a_n denotes the nth term of the A.P. 3, 8, 13, 18, ..., then the value of a₃₀ – a₂₀ is ______.

In an A.P. a₁, a₂, a₃,...,a_n,...., if a₁ = 1, a_n = 20 and S_n = 399, then the value of n is ______.

If 7 times the 7^th term of an A.P. is equal to 11 times its 11^th term, then the value of its 18^th term is ______.

Two arithmetic progressions have the same common difference. Their first terms are A and B respectively. The difference between their n^th terms is ______.

If the n^th terms of two A.P.s: 9, 7,5 .... and 24, 21, 18, ... are the same, then the value of n is ______.

If S_n = n (4n + 1) is the sum of n terms of an A.P., then its common difference is ______.

If the ratio of the sums of first n terms of two A.P.s is `(5n + 13)/(7n + 27)`, then the ratio of their 4^th terms is ______.