NCERT solutions for Mathematics [English] Class 11 chapter 8 - Sequences and Series [Latest edition]

Write the first five terms of the sequences whose n^th term is:

`a_n = n/(n + 1)`

Write the first five terms of the sequences whose n^th term is:

a_n = 2ⁿ

Write the first five terms of the sequences whose n^th term is:

`a_n = (2n -3)/6`

Write the first five terms of the sequences whose n^th term is:

`a_n = (-1)^(n-1) 5^(n+1)`

Write the first five terms of the sequences whose n^th term is:

`a_n = n (n^2 + 5)/4`

Find the indicated term in the following sequence whose n^th term is:

a_n = 4n – 3; a_17^, a₂₄ 

Find the indicated term in the following sequence whose n^th term is:

`a_n = n^2/2^n`; `a_7`

Find the indicated term in the following sequence whose n^th term is:

`a_n = (–1)^(n – 1) n^3; a_9`

Find the indicated term in the following sequence whose n^th term is:

`a_n = (n(n-2))/(n+3)` ;`a_20`

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

a₁ = 3, a_n = 3a_{(n - 1)} + 2 for all n > 1

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`

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`

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

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

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

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

Which term of the following sequence: 

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

Which term of the following sequence:

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

Which term of the following sequence:

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

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

Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.

Find the sum to indicated number of terms in the geometric progressions 1, – a, a², – a³, ... n terms (if a ≠ – 1).

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

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

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.

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.

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

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`

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

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 4^th by 18.

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`

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

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

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.

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

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))`.

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 2^nd hour, 4^th hour and n^th hour? 

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

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

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.

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.

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.

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.

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.

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

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

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

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

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

5 + 55 + 555 + …

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

.6 +.66 +. 666 +…

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?

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?

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 8^th set of letter is mailed.

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

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.

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.