NCERT Solutions

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are:

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are: 2. $a_{n}=\frac{n}{n+1}$

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are: 3. $a_{n}=2^{n}$

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are: 4. $a_{n}=\frac{2 n-3}{6}$

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are: 5. $a_{n}=(-1)^{n-1} 5^{n+1}$

Write the first five terms of each of the sequences in Exercises 1 to 6 whose $n^{\text {th }}$ terms are: 6. $a_{n}=n \frac{n^{2}+5}{4}$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose $n^{\text {th }}$ terms are: 7. $a_{n}=4 n-3 ; a_{17}, a_{24}$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose $n^{\text {th }}$ terms are: 8. $a_{n}=\frac{n^{2}}{2^{n}} ; a_{7}$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose $n^{\text {th }}$ terms are: 9. $a_{n}=(-1)^{n-1} n^{3} ; a_{9}$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose $n^{\text {th }}$ terms are: 10. $a_{n}=\frac{n(n-2)}{n+3} ; a_{20}$

Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: 11. $a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n>1$

Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: 12. $a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2$

Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: 13. $a_{1}=a_{2}=2, a_{n}=a_{n-1}-1, n>2$

The Fibonacci sequence is defined by $1=a_{1}=a_{2}$ and $a_{n}=a_{n-1}+a_{n-2}, n>2$

Find $\frac{a_{n+1}}{a_{n}}$, for $n=1,2,3,4,5$

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

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 $q^{2}=p s$.

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

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

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 7. $0.15,0.015,0.0015, \ldots 20$ terms.

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 8. $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, \ldots n$ terms.

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 9. $1,-a, a^{2},-a^{3}, \ldots n$ terms (if $a \neq-1$ ).

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 10. $x^{3}, x^{5}, x^{7}, \ldots n$ terms (if $x \neq \pm 1$ ).

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

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

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

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.

Given a G.P. with $a=729$ and $7^{\text {th }}$ term 64, determine $\mathrm{S}_{7}$.

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

If the $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ 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 $128,32,8,2, \frac{1}{2}$.

Show that the products of the corresponding terms of the sequences $a, a r, a r^{2}, \ldots a r^{n-1}$ and $\mathrm{A}, \mathrm{AR}, \mathrm{AR}^{2}, \ldots \mathrm{AR}^{n-1}$ 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 $4^{\text {th }}$ 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 $\mathrm{P}^{2}=(a b)^{n}$.

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

If $a, b, c$ and $d$ are in G.P. show that $\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2}$.

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

Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ 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 $(3+2 \sqrt{2}):(3-2 \sqrt{2})$.

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

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 $2^{\text {nd }}$ hour, $4^{\text {th }}$ hour and $n^{\text {th }}$ 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 $x, y \in \mathbf{N}$ 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 $\frac{a+b x}{a-b x}=\frac{b+c x}{b-c x}=\frac{c+d x}{c-d x}(x \neq 0)$, 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 $\mathrm{P}^{2} \mathrm{R}^{n}=\mathrm{S}^{n}$.

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

If $a$ and $b$ are the roots of $x^{2}-3 x+p=0$ and $c, d$ are roots of $x^{2}-12 x+q=0$, 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 $a: b=\left(m+\sqrt{m^{2}-n^{2}}\right):\left(m-\sqrt{m^{2}-n^{2}}\right)$.

Find the $20^{\text {th }}$ 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 $8^{\text {th }}$ 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 $15^{\text {th }}$ 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.