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NCERT solutions Mathematics Class 11 chapter 9 Sequences and Series

Chapters

NCERT Mathematics Class 11

Mathematics Textbook for Class 11

Chapter 9 - Sequences and Series

Pages 180 - 181

Write the first five terms of the sequences whose nth term is `a_n = n(n+2)`

 
Q 1 | Page 180

Write the first five terms of the sequences whose nth term is  `a_n = n/(n + 1)`

Q 2 | Page 180

Write the first five terms of the sequences whose nth term is an = 2n

Q 3 | Page 180

Write the first five terms of the sequences whose nth term is  `a_n = (2n -3)/6`

Q 4 | Page 180

Write the first five terms of the sequences whose nth term is  `a_n = (-1)^(n-1) 5^(n+1)`

Q 5 | Page 180

Write the first five terms of the sequences whose nth term is  `a_n = n (n^2 + 5)/4`

Q 6 | Page 180

Find the 17th term in the following sequence whose nth term is an = 4n – 3; `a_17 , a_24`

Q 7 | Page 180

Find the indicated terms in each of the sequences whose nth terms are `a_n = n^2/2^n`; `a_7`

Q 8 | Page 180

Find the 9th term in the following sequence whose nth term is `a_n = (–1)^(n – 1) n^3; a_9`

Q 9 | Page 180

Find the 20th term in the following sequence whose nth term is `a_n = (n(n-2))/(n+3)` ;`a_20`

Q 10 | Page 180

Write the first five terms of the following sequence and obtain the corresponding series:

`a_1 = 3, a_n = 3a_( n-1) + 2 for all n > 1`

Q 11 | Page 181

Write the first five terms of the following sequence and obtain the corresponding series:   `a_1 = -1, a_n = (a_(n-1))/n , n >= 2`

Q 12 | Page 181

Write the first five terms of the following sequence and obtain the corresponding series: 

`a_1 = a_2 = 2, a_n = a_(n-1) -1, n > 2`

Q 13 | Page 181

The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2 , n > 2.

Find `a_(n+1)/a_n`, for n = 1, 2, 3, 4, 5

 

Q 14 | Page 181

Pages 185 - 186

Find the sum of odd integers from 1 to 2001.

Q 1 | Page 185

Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

Q 2 | Page 185

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.

Q 3 | Page 185

How many terms of the A.P.  -6 , `-11/2` , -5... are needed to give the sum –25?

Q 4 | Page 185

In an A.P., if pth term is 1/q and qth term is 1/p,  prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`

Q 5 | Page 185

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

Q 6 | Page 185

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

Q 7 | Page 185

If the sum of n terms of an A.P. is (pn qn2), where p and q are constants, find the common difference.

Q 8 | Page 185

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

Q 9 | Page 185

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Q 10 | Page 185

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`

Q 11 | Page 185

The ratio of the sums of m and n terms of an A.P. is m2n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)

Q 12 | Page 185

If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.

Q 13 | Page 185

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

Q 14 | Page 185

if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.

Q 15 | Page 185

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.

Q 16 | Page 185

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

Q 17 | Page 186

The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.

Q 18 | Page 186

Pages 192 - 193

Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`

Q 1 | Page 192

Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.

Q 2 | Page 192

The 5th, 8th and 11th terms of a G.P. are pq and s, respectively. Show that q2 = ps.

Q 3 | Page 192

The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.

Q 4 | Page 192

Which term of the following sequences: 

`2, 2sqrt2, 4,.... is 128`

Q 5.1 | Page 192

Which term of the following sequences:

`sqrt3, 3, 3sqrt3, .... is 729`?

Q 5.2 | Page 192

Which term of the following sequences:

`1/3, 1/9, 1/27, .... is 1/19683` ?

Q 5.3 | Page 192

For what values of x, the numbers  `2/7, x, -7/2` are in G.P?

Q 6 | Page 192

Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 …

Q 7 | Page 192

Find the sum to n terms in the geometric progression `sqrt7, sqrt21,3sqrt7...`

Q 8 | Page 192

Find the sum to n terms in the geometric progression 1, – a, a2, – a3, ... n terms (if a ≠ – 1).

Q 9 | Page 192

Find the sum to n terms in the geometric progression x3, x5, x7, ... n terms (if x ≠ ± 1).

Q 10 | Page 192

Evaluate `sum_(k=1)^11 (2+3^k )`

Q 11 | Page 192

The sum of first three terms of a G.P. is  `39/10` and their product is 1. Find the common ratio and the terms.

Q 12 | Page 192

How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?

Q 13 | Page 192

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

Q 14 | Page 192

Given a G.P. with a = 729 and 7th term 64, determine S7.

Q 15 | Page 192

Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

Q 16 | Page 192

If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.

Q 17 | Page 192

Find the sum to n terms of the sequence, 8, 88, 888, 8888…

Q 18 | Page 193

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2

Q 19 | Page 193

Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio

Q 20 | Page 193

Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.

Q 21 | Page 193

If the `p^(th), q^(th) and r^(th)` terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`

Q 22 | Page 193

If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of terms, prove that P2 = (ab)n.

Q 23 | Page 193

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from `(n + 1)^(th) " to "(2n)^(th) " term is " 1/r^n`.

 
Q 24 | Page 193

If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .

Q 25 | Page 193

Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Q 26 | Page 193

Find the value of n so that  `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.

Q 27 | Page 193

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `3 + 2sqrt2) ":" (3 - 2sqrt2)`

Q 28 | Page 193

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are `A+- sqrt((A+G)(A-G))`

Q 29 | Page 193

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nthhour?

Q 30 | Page 193

What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

Q 31 | Page 193

If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

Q 32 | Page 193

Page 196

Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + …

Q 1 | Page 196

Find the sum to n terms of the series 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + …

Q 2 | Page 196

Find the sum to n terms of the series 3 × 12 + 5 × 22 + 7 × 32 + …

Q 3 | Page 196

Find the sum to n terms of the series `1/(1xx2) + 1/(2xx3)+1/(3xx4)+ ...`

Q 4 | Page 196

Find the sum to n terms of the series  52 + 62 + 72 + ... + 202

Q 5 | Page 196

Find the sum to n terms of the series 3 × 8 + 6 × 11 + 9 × 14 +…

Q 6 | Page 196

Find the sum to n terms of the series 12 + (12 + 22) + (12 + 22 + 32) + …

Q 7 | Page 196

Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4).

Q 8 | Page 196

Find the sum to n terms of the series whose nth terms is given by n2 + 2n

Q 9 | Page 196

Find the sum to n terms of the series whose nth terms is given by (2n – 1)2

Q 10 | Page 196

Pages 199 - 200

Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

Q 1 | Page 199

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

Q 2 | Page 199

Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)

Q 3 | Page 199

Find the sum of all numbers between 200 and 400 which are divisible by 7.

Q 4 | Page 199

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

Q 5 | Page 199

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

Q 6 | Page 199

If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.

Q 7 | Page 199

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

Q 8 | Page 199

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Q 9 | Page 199

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Q 10 | Page 199

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Q 11 | Page 199

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Q 12 | Page 199

if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that abc and d are in G.P.

Q 13 | Page 199

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn

Q 14 | Page 199

The pthqth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0

Q 15 | Page 199

if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.

Q 16 | Page 199

If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.

Q 17 | Page 199

If a and are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, cd, form a G.P. Prove that (q + p): (q – p) = 17:15.

Q 18 | Page 199

The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that `a:b = (m + sqrt(n^2 - n^2)):(m - sqrt(m^2 - n^2))`.

Q 19 | Page 200

If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.

 
Q 20 | Page 200

Find the sum of the following series up to n terms:

5 + 55 + 555 + …

Q 21.1 | Page 200

Find the sum of the following series up to n terms:

.6 +.66 +. 666 +…

Q 21.2 | Page 200

Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + … + n terms.

Q 22 | Page 200

Find the sum of the first n terms of the series: 3 + 7 + 13 + 21 + 31 + …

Q 23 | Page 200

If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that  `9S_2^2 = S_3(1 + 8S_1)`

Q 24 | Page 200

Find the sum of the following series up to n terms `1^3/1 + (1^3 + 2^3)/(1+3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) +...`

Q 25 | Page 200

Show that  `(1xx2^2 + 2xx3^2 + ...+nxx(n+1)^2)/(1^2 xx 2 + 2^2 xx3 + ... + n^2xx (n+1))` = `(3n + 5)/(3n + 1)`

Q 26 | Page 200

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

Q 27 | Page 200

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

Q 28 | Page 200

A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

Q 29 | Page 200

A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

Q 30 | Page 200

A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Q 31 | Page 200

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

Q 32 | Page 200

NCERT Mathematics Class 11

Mathematics Textbook for Class 11
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