NCERT Solutions
Class 11 - Mathematics

Chapter 8: SEQUENCES AND SERIES

Write the first five terms of the sequence whose nth term is \(a_n = n(n+2)\).

Write the first five terms of the sequence whose nth term is \(a_n = \dfrac{n}{n+1}\).

Write the first five terms of the sequence whose nth term is \(a_n = 2^n\).

Write the first five terms of the sequence whose nth term is \(a_n = \dfrac{2n - 3}{6}\).

Write the first five terms of the sequence whose nth term is \(a_n = (-1)^{n-1} 5^{n+1}\).

Write the first five terms of the sequence whose nth term is \(a_n = n\left(\dfrac{n^2 + 5}{4}\right)\).

Find the indicated terms of the sequence whose nth term is \(a_n = 4n - 3\): Find \(a_{17}\) and \(a_{24}\).

Find the indicated term for the sequence whose nth term is \(a_n = \dfrac{n^2}{2^n}\): Find \(a_7\).

Find the indicated term of the sequence whose nth term is \(a_n = (-1)^n 1^n 3^n\): Find \(a_9\).

Find the indicated term of the sequence whose nth term is \(a_n = \dfrac{n(n - 2)}{n + 3}\): Find \(a_{20}\).

Write the first five terms of the recursively defined sequence: \(a_1 = 3\), \(a_n = 3a_{n-1} + 2\) for \(n > 1\). Also write the corresponding series.

Write the first five terms of the sequence defined by \(a_1 = -1\) and \(a_n = \dfrac{a_{n-1}}{n}\) for \(n \geq 2\). Also write the corresponding series.

Write the first five terms of the sequence defined by \(a_1 = a_2 = 2\) and \(a_n = a_{n-1} - 1\) for \(n > 2\). Also write the corresponding series.

The Fibonacci sequence is defined by \(a_1 = a_2 = 1\) and \(a_n = a_{n-1} + a_{n-2}\) for \(n > 2\). Find \(\dfrac{a_{n+1}}{a_n}\) for \(n = 1, 2, 3, 4, 5\).

Find the \(20^{\text{th}}\) and \(n^{\text{th}}\) terms of the G.P. \( \dfrac{5}{2}, \dfrac{5}{4}, \dfrac{5}{8}, \ldots \).

Find the \(12^{\text{th}}\) term of a G.P. whose \(8^{\text{th}}\) term is \(192\) and the common ratio is \(2\).

The \(5^{\text{th}}\), \(8^{\text{th}}\) and \(11^{\text{th}}\) terms of a G.P. are \(p\), \(q\) and \(s\), respectively. Show that \(q^{2} = ps\).

The \(4^{\text{th}}\) term of a G.P. is the square of its second term, and the first term is \(-3\). Determine its \(7^{\text{th}}\) term.

Which term of each of the following sequences has the indicated value?

(a) In the sequence \(2, 2\sqrt{2}, 4, \ldots\), which term is \(128\)?

(b) In the sequence \(\sqrt{3}, 3, 3\sqrt{3}, \ldots\), which term is \(729\)?

(c) In the sequence \( \dfrac{1}{3}, \dfrac{1}{9}, \dfrac{1}{27}, \ldots \), which term is \( \dfrac{1}{19683} \)?

For what values of \(x\) are the numbers \(-\dfrac{2}{7},\ x,\ \dfrac{7}{2}\) in G.P.?

Find the sum to the indicated number of terms in the geometric progression.

\(0.15,\ 0.015,\ 0.0015, \ldots\) \(20\) terms.

Find the sum to \(n\) terms of the geometric progression.

\(\sqrt{7},\ \sqrt{21},\ 3\sqrt{7}, \ldots,\ n\text{ terms}.\)

Find the sum to \(n\) terms of the geometric progression \(1, -a, a^{2}, -a^{3}, \ldots\) (if \(a \neq -1\)).

Find the sum to \(n\) terms of the geometric progression \(x^{3}, x^{5}, x^{7}, \ldots\) (if \(x \neq \pm 1\)).

Evaluate \(\displaystyle \sum_{k=1}^{11} (2 + 3^{k})\).

The sum of the first three terms of a G.P. is \(\dfrac{39}{10}\) and their product is \(1\). Find the common ratio and the three terms.

How many terms of the G.P. \(3, 3^{2}, 3^{3}, \ldots\) are needed to give the sum \(120\)?

The sum of the 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.

Given a G.P. with \(a = 729\) and \(7^{\text{th}}\) term \(64\), determine \(S_{7}\).

Find a G.P. for which the sum of the first two terms is \(-4\) and the fifth term is four times the third term.

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.

Find the sum to \(n\) terms of the sequence \(8, 88, 888, 8888, \ldots\).

Find the sum of the products of the corresponding terms of the sequences \(2, 4, 8, 16, 32\) and \(128, 32, 8, 2, \tfrac{1}{2}\).

Show that the products of the corresponding terms of the sequences \(a, ar, ar^{2}, \ldots, ar^{n-1}\) and \(A, AR, AR^{2}, \ldots, AR^{n-1}\) form a G.P., and find the common ratio.

Find four numbers forming a geometric progression in which the third term is greater than the first term by \(9\), and the second term is greater than the fourth term by \(18\).

If the \(p^{\text{th}}\), \(q^{\text{th}}\) and \(r^{\text{th}}\) 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 \(n^{\text{th}}\) term of a G.P. are \(a\) and \(b\), respectively, and if \(P\) is the product of \(n\) terms, prove that \(P^{2} = (ab)^{n}\).

Show that the ratio of the sum of the first \(n\) terms of a G.P. to the sum of the terms from \((n+1)^{\text{th}}\) to \((2n)^{\text{th}}\) term is \(\dfrac{1}{r^{n}}\), where \(r\) is the common ratio.

If \(a, b, c\) and \(d\) are in G.P., show that

\((a^{2} + b^{2} + c^{2})(b^{2} + c^{2} + d^{2}) = (ab + bc + cd)^{2}.\)

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

Find the value of \(n\) so that \( \dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}} \) may be the geometric mean between \(a\) and \(b\).

The sum of two numbers is six times their geometric mean. Show that the numbers are in the ratio \((3 + 2\sqrt{2}) : (3 - 2\sqrt{2})\).

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

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 2nd hour, 4th hour and \(n^{\text{th}}\) hour?

What will Rs \(500\) amount to in 10 years after its deposit in a bank which pays an annual interest rate of \(10\%\) compounded annually?

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

If \(f\) is a function satisfying \(f(x + y) = f(x) f(y)\) for all \(x, y \in \mathbb{N}\) such that \(f(1) = 3\) and \(\displaystyle \sum_{x = 1}^{n} f(x) = 120\), find the value of \(n\).

The sum of some terms of a 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 the 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 \(\dfrac{a + bx}{a - bx} = \dfrac{b + cx}{b - cx} = \dfrac{c + dx}{c - dx}\) (\(x \neq 0\)), 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 \(P^{2} R^{n} = S^{n}\).

If \(a, b, c, d\) are in G.P., prove that \((a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d^{n})\) are in G.P.

If \(a\) and \(b\) are the roots of \(x^{2} - 3x + p = 0\) and \(c\) and \(d\) are roots of \(x^{2} - 12x + q = 0\), where \(a, b, c, d\) form a G.P., prove that \((q + p) : (q - p) = 17 : 15\).

The ratio of the A.M. and G.M. of two positive numbers \(a\) and \(b\) is \(m : n\). Show that

\[a : b = \bigl(m + \sqrt{m^{2} - n^{2}}\bigr) : \bigl(m - \sqrt{m^{2} - n^{2}}\bigr).\]

Find the sum of the following series up to \(n\) terms:

(i) \(5 + 55 + 555 + \ldots\)

(ii) \(0.6 + 0.66 + 0.666 + \ldots\)

Find the 20th term of the series \(2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots\) \(n\) terms.

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 will 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 instalment 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 8th 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 15th 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 the second day, 4 more workers dropped out on the 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.