SEQUENCES AND SERIES - 6 (Sequences and Series - Problem Solving)

Sequence

A sequence is a function of natural numbers with codomain as the set of real numbers. It is said to be finite or infinite according it has finite or infinite number of terms. Sequence $a _{1}, a _{2}, \ldots \ldots a _{n}$ is usually denoted by $\left\{a _{n}\right\}$ or $<a _{n}>$

Series

By adding or subtracting the terms of a sequence we get a series.

Arithmetic Progression (A.P.)

It is a sequence in which the difference between two consecutive terms is the same.

For a sequence $\left\{a _{n}\right\}$ which is in A.P, $n^{\text {th }}$ term $a _{n}=a+(n-1) d=\ell$ (last term) which is always a linear expression in $\mathrm{n}$ )

$\mathrm{d}=\mathrm{a} _{\mathrm{n}}-\mathrm{a} _{\mathrm{n}-1}$ (If $\mathrm{d}=0$, then sequence is a constant sequence. if $\mathrm{d}>0$ the sequence is increasing; if $\mathrm{d}<0$, the sequence is decreasing)

$\mathrm{n}^{\text {th }}$ term from the end $\mathrm{a} _{\mathrm{n}}{ }^{1}=\ell+(\mathrm{n}-1)(-\mathrm{d})$

$ =\ell-(\mathrm{n}-1) \mathrm{d} $

Sum to $\mathrm{n}$ terms $=\left\{\begin{array}{c}\frac{\mathrm{n}}{2}(2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}) \\ \text { or } \\ =\frac{\mathrm{n}}{2}(\mathrm{a}+\ell)\end{array}\right.$

$\left(\mathrm{S} _{\mathrm{n}}\right.$ is a quadratic expression in $\mathrm{n}$; common difference $=\frac{1}{2}$ coefficient of $\left.\mathrm{n}^{2}\right)$

Also $\mathrm{a} _{\mathrm{n}}=\mathrm{S} _{\mathrm{n}}-\mathrm{S} _{\mathrm{n}-1}$

Arithmetic mean

If $a, b, c$ are in $A . P$, then $b=\frac{a+c}{2}$ is called the single arithmetic mean of $a \& c$. Let $a \& b$ be two given numbers and $A _{1}, A _{2}, \ldots \ldots \ldots . . . . A _{n}$ are $n$ A.M’s between them. Then $a, A _{1}, A _{2}, \ldots A _{n}, b$ are in A.P. Common difference of this sequence $\mathrm{d}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}+1}$.

$\mathrm{A} _{1}=\mathrm{a}+\mathrm{d}, \mathrm{A} _{2}=\mathrm{a}+2 \mathrm{~d}$ etc. we can find all the arithmetic means.

Properties of A.P.

1. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots \ldots .$. are in A.P; then $\mathrm{a}_1 \pm \mathrm{k}, \mathrm{a}_2 \pm \mathrm{k}, \mathrm{a}_3 \pm \mathrm{k}$, $…….$ are also in A.P.

2. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3$ $………$ are in A.P, then $\mathrm{a}_1 \lambda, \mathrm{a}_2 \lambda, \mathrm{a}_3 \lambda$ $………..$ and $\frac{\mathrm{a}_1}{\lambda}, \frac{\mathrm{a}_2}{\lambda}, \frac{\mathrm{a}_3}{\lambda}$ are also in A.P $\left(\lambda^{\prime} \neq 0\right)$

3. If $ a_1, {a}_2, \ldots \ldots . . a_n $ are in A.P, then $a_n,a_n-1, \ldots \ldots \ldots \ldots . . a_2, a_1$ is also an A.P with common difference ( $-$d)

4. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3$, $……….$ and $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3$, $……..$ are two A.P.s then $\mathrm{a}_1 \pm \mathrm{b}_1, \mathrm{a}_2 \pm \mathrm{b}_2, \mathrm{a}_3 \pm \mathrm{b}_3, \ldots$. are also in A.P.

5. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots \ldots \ldots \ldots \ldots$ and $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \ldots \ldots \ldots \ldots \ldots \ldots$ are two A.P.s then $\mathrm{a}_1 \mathrm{~b}_1, \mathrm{a}_2 \mathrm{~b}_2, \mathrm{a}_3 \mathrm{~b}_3, \ldots \ldots \ldots \ldots$ and $\frac{\mathrm{a}_1}{\mathrm{~b}_1}, \frac{\mathrm{a}_2}{\mathrm{~b}_2}, \frac{\mathrm{a}_3}{\mathrm{~b}_3}, \ldots \ldots \ldots . . . . . . . \mathrm{are}$ NOT in A.P.

6. If 3 numbers are in A.P we may take them as $a-d, a, a+d$. If 4 numbers are in A.P, we can take them as $a-3 \mathrm{~d}, \mathrm{a}-\mathrm{d}, \mathrm{a}+\mathrm{d}, \mathrm{a}+3 \mathrm{~d}$.

7. In an arithmetic progression, sum of the terms equidistant form the beginning and end is a constant and equal to sum of first and last term.

$\quad$ ie for $\left\{a _{n}\right\}$,

$\quad$ $\mathrm{a} _{1}+\mathrm{a} _{\mathrm{n}}=\mathrm{a} _{2}+\mathrm{a} _{\mathrm{n}-1}=\mathrm{a} _{3}+\mathrm{a} _{\mathrm{n}-2}=\ldots \ldots$

$\quad$ Also $\mathrm{a} _{\mathrm{r}}=\frac{\mathrm{a} _{\mathrm{r}-\mathrm{k}}+\mathrm{a} _{\mathrm{r}+\mathrm{k}}}{2}, 0 \leq \mathrm{k} \leq \mathrm{n}-\mathrm{r}$.

8. Sum of $n$ arithmetic means between two given numbers a & $b$ is $n$ times the single A.M between them.

$\quad$ ie. $A _{1}+A _{2}+\ldots \ldots \ldots \ldots . .+A _{n}=n\left(\frac{a+b}{2}\right)$

9. Also $S _{n}=a _{1}+a _{2}+\ldots \ldots+a _{n}=\left\{\begin{array}{l}n(\text { middle term }) ; \text { if } n \text { is odd. } \\ \frac{n}{2} \text { (sum of two middle terms); if } n \text { is even }\end{array}\right.$

Geometric Progression (G.P.)

It is a sequence in which the ratio of any two consecutive terms is the same. For a sequence $\left\{a _{n}\right\}$ which is in G.P. $\mathrm{n}^{\text {th }}$ term $\mathrm{a} _{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ (last term)

Common ratio $r=\frac{a _{n}}{a _{n-1}}\left(r _{\neq} \neq 0\right.$. If $r>1$, the sequence is an increasing sequence, if $0<r<1$ then the sequence is decreasing )

$n^{\text {th }}$ term from the end $a _{n}{ }^{1}=a _{n}\left(\frac{1}{r}\right)^{n-1} \quad\left(a _{n}{ }^{1}=n\right.$th term from end $)$

Note : No term of G.P. can be zero

Sum to $n$ terms $S _{n}=\left\{\begin{array}{l}\frac{a\left(r^{n}-1\right)}{r-1}, r \neq 1 \\ n a, r=1\end{array}\right.$

If $|r|<1$, the sum of the infinite G.P is given by $S _{\infty}=\frac{a}{1-r}$

Geometric mean

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in G.P, then $\mathrm{b}^{2}=\mathrm{ac}$ or $\mathrm{b}=\sqrt{\mathrm{ac}}$ is called the single geometric mean of $\mathrm{a} \& \mathrm{c}$. Let $\mathrm{a} \&$ $\mathrm{b}$ be two given numbers and $\mathrm{G} _{1}, \mathrm{G} _{2}, \ldots . . \mathrm{G} _{\mathrm{n}}$ are $\mathrm{n}$ G.M.s between them. Then $\mathrm{a}, \mathrm{G} _{1}, \mathrm{G} _{2}, \ldots \ldots . . \mathrm{G} _{\mathrm{n}}$,

$b$ are in G.P. Common ratio of this sequence $r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

$\mathrm{G} _{1}=\mathrm{ar}, \mathrm{G} _{2}=\mathrm{ar}^{2}$ etc. we can find all the geometric means.

Properties of G.P.

1. If $a_1, a_2, a_3$ $……$ are in G.P., then $ a_1 k , a_2 k, a_3 k$, $……$ and $\frac{ a_1}{ k}, \frac{ a_2}{ k}, \frac{ a_3}{ k}$,$………$are also in G.P $\left( {k}_{\neq 0}\right)$.

2. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3$, $………$ are in G.P., then $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$, $………..$ and $\mathrm{a}_1{ }^{\mathrm{n}}, \mathrm{a}_2{ }^{\mathrm{n}}, \mathrm{a}_3{ }^{\mathrm{n}}$, $………..$ are also in G.P.

3. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}, \ldots \ldots . . \mathrm{a} _{\mathrm{n}}$ are in G.P with common ratio $\mathrm{r}$, then $\mathrm{a} _{\mathrm{n}}, \mathrm{a} _{\mathrm{n}-1} \ldots \ldots \ldots \ldots . \mathrm{a} _{2}, \mathrm{a} _{1}$ is also in G.P. With common ratio $\frac{1}{\mathrm{r}}$. G.P.

4. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots \ldots .$. and $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \ldots \ldots . .$. are two G.P.s then $\mathrm{a}_1 \pm \mathrm{b}_1, \mathrm{a}_2 \pm \mathrm{b}_2, \mathrm{a}_3 \pm \mathrm{b}_3, \ldots \ldots .$. are NOT in G.P.

5. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3$, $……….$ and $b_1, b_2, b_3$ $……..$ are two G.P.s then $\mathrm{a}_1 \mathrm{~b}_1, \mathrm{a}_2 \mathrm{~b}_2, \mathrm{a}_3 \mathrm{~b}_3$ $………$ and $\frac{\mathrm{a}_1}{\mathrm{~b}_1}, \frac{\mathrm{a}_2}{\mathrm{~b}_2}, \frac{\mathrm{a}_3}{\mathrm{~b}_3}$ are also in G.P.

6. If 3 numbers are in G.P., we may take them as $\frac{\mathrm{a}}{\mathrm{r}}$, a, ar. If 4 numbers are in G.P., we can take them as $\frac{\mathrm{a}}{\mathrm{r}^{3}}, \frac{\mathrm{a}}{\mathrm{r}}$, ar, a $\mathrm{r}^{3}$.

7. In a geometric progression, product of the terms equidistant from the beginning and end is a constant and equal to product of first and last term.

ie $\quad$ For $\left\{\mathrm{a} _{\mathrm{n}}\right\}$

$\mathrm{a} _{1} \mathrm{a} _{\mathrm{n}}=\mathrm{a} _{2} \mathrm{a} _{\mathrm{n}-1}=\mathrm{a} _{3} \mathrm{a} _{\mathrm{n}-2}=$

Also $\mathrm{a} _{\mathrm{r}}=\sqrt{\mathrm{a} _{\mathrm{r}-\mathrm{k}} \mathrm{a} _{\mathrm{r}+\mathrm{k}}}, 0 \leq \mathrm{k} \leq \mathrm{n}-\mathrm{r}$.

8. Product of $n$ geometric means between two given numbers $a \& b$ is $n^{\text {th }}$ power of the single G.M. between them.

ie $\mathrm{G} _{1} \mathrm{G} _{2} \mathrm{G} _{3} \ldots \ldots \ldots \mathrm{G} _{\mathrm{n}}=(\sqrt{\mathrm{ab}})^{\mathrm{n}}$

9. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}, \ldots \ldots \ldots \ldots \ldots . . .$. are in G.P. $\left(\mathrm{a} _{\mathrm{i}}>0 \forall \mathrm{i}\right)$, then $\log \mathrm{a} _{1}, \log \mathrm{a} _{2}, \log \mathrm{a} _{3}, \ldots \ldots$ are in A.P. Its converse is also true.

Harmonic Progression (H.P.)

A sequence is said to be in H.P if the reciprocals of its terms are in A.P.

ie. if $a _{1}, a _{2}, a _{3}, \ldots \ldots . a _{n}$ are in H.P., then $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \ldots . . \frac{1}{a _{n}}$ are in A.P.

For a sequence $\left\{\mathrm{a} _{\mathrm{n}}\right\}$ which is in H.P.,

$n^{\text {th }}$ term $a _{n}=\frac{1}{\frac{1}{a _{1}}+(n-1)\left(\frac{1}{a _{2}}-\frac{1}{a _{1}}\right)}=\frac{a _{1} a _{2}}{a _{2}+(n-1)\left(a _{1}-a _{2}\right)}$

$n^{\text {th }}$ term from end $\mathrm{a} _{\mathrm{n}}{ }^{1}=\frac{1}{\frac{1}{\mathrm{a} _{\mathrm{n}}}-(\mathrm{n}-1)\left(\frac{1}{\mathrm{a} _{2}}-\frac{1}{\mathrm{a} _{1}}\right)}=\frac{\mathrm{a} _{1} \mathrm{a} _{2} \mathrm{a} _{\mathrm{n}}}{\mathrm{a} _{1} \mathrm{a} _{2}-\mathrm{a} _{\mathrm{n}}(\mathrm{n}-1)\left(\mathrm{a} _{1}-\mathrm{a} _{2}\right)}$

Note: No term of H.P. can be zero. There is no general formula for finding out the sum of $n$ terms of H.P.

Harmonic mean

If $a, b, c$ are in H.P; then $b=\frac{2 a c}{a+c}$ is called the single H.M. between $a \& c$. Let $a \& b$ be two given numbers and $\mathrm{H} _{1}, \mathrm{H} _{2}, \ldots \ldots \ldots \ldots . ., \mathrm{H} _{\mathrm{n}}$ are $\mathrm{n}$ H.M.s between them. then $\mathrm{a}, \mathrm{H} _{1}, \mathrm{H} _{2}, \ldots \ldots . \mathrm{H} _{\mathrm{n}}$, b are in H.P. The common difference $d$ of the corresponding A.P is

$\mathrm{d}=\frac{\mathrm{a}-\mathrm{b}}{(\mathrm{n}+1) \mathrm{ab}}$

$\frac{1}{\mathrm{H} _{1}}=\frac{1}{\mathrm{a}}+\mathrm{d}, \frac{1}{\mathrm{H} _{2}}=\frac{1}{\mathrm{a}}+2 \mathrm{~d}$ etc. we can find all the harmonic means.

Note: The sum of reciprocals of $n$ Harmonic means between two given numbers is $n$ times the reciprocal of single H.M. between them.

ie $\quad \frac{1}{\mathrm{H} _{1}}+\frac{1}{\mathrm{H} _{2}}+\ldots \ldots \frac{1}{\mathrm{H} _{\mathrm{n}}}=\mathrm{n} \frac{\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}\right)}{2}$

Note: If $a, b, c$ are three successive terms of a sequence. Then

$\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}-\mathrm{c}}=\left\{\begin{array}{l}\frac{\mathrm{a}}{\mathrm{a}} \Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in A.P. } \\ \frac{\mathrm{a}}{\mathrm{b}} \Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in G.P. } \\ \frac{\mathrm{a}}{\mathrm{c}} \Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in H.P. }\end{array}\right.$

Relation between A.M., G.M., and H.M.

For positive numbers $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}, \ldots \ldots \ldots \ldots \ldots . \mathrm{a} _{\mathrm{n}}$

A.M. $=A=\frac{\mathrm{a} _{1}+\mathrm{a} _{2}+\ldots \ldots .+\mathrm{a} _{\mathrm{n}}}{\mathrm{n}}$

G.M. $=\mathrm{G}=\left(\mathrm{a} _{1} \mathrm{a} _{2} \ldots \ldots . \mathrm{a} _{\mathrm{n}}\right)^{\frac{1}{\mathrm{n}}}$

$H . M=H=\frac{n}{\frac{1}{a _{1}}+\frac{1}{a _{2}}+\ldots . .+\frac{1}{a _{n}}}$,

$\mathrm{A} \geq \mathrm{G} \geq \mathrm{H}$ and $\mathrm{G}^{2}=\mathrm{AH}$.

(equality holds if $\mathrm{a} _{1}=\mathrm{a} _{2}=$. $ \qquad $ Note : Also $\sqrt{\frac{\mathrm{a} _{1}{ }^{2}+\mathrm{a} _{2}{ }^{2}+\ldots \ldots .+\mathrm{a} _{\mathrm{n}}{ }^{2}}{\mathrm{n}}} \geq \frac{\mathrm{a} _{1}+\mathrm{a} _{2}+\ldots \ldots . .+\mathrm{a} _{\mathrm{n}}}{\mathrm{n}}$

(Root mean square inequality)

Note: The quadratic equation having $a, b$ as its roots is $x^{2}-2 A x+G^{2}=0$ and $a: b=A+\sqrt{A^{2}-G^{2}}$ :A- $\sqrt{\mathrm{A}^{2}-\mathrm{G}^{2}}$ where $\mathrm{A}, \mathrm{G}$ are respectively the A.M. and G.M. of $\mathrm{a} \& \mathrm{~b}$

Note : Formation of progressions

Two consecutive terms determine the required progression. If two numbers $\mathrm{a} \& \mathrm{~b}$ are given, then

(i) $\quad \mathrm{a}, \mathrm{b}, 2 \mathrm{~b}-\mathrm{a}$ is A.P.

(ii) $\quad a, b, \frac{b^{2}}{\mathrm{a}}$ is G.P.

(ii) $\quad a, b,$ $\frac{\mathrm{ab}}{2 \mathrm{a}-\mathrm{b}}$ is H.P.

Solved examples

1. If the $\mathrm{p}^{\text {th }}, \mathrm{q}^{\text {th }}$ and $\mathrm{r}^{\text {th }}$ terms of an A.P are in GP, then the common ratio of the G.P is

(a) $\frac{\mathrm{p}+\mathrm{q}}{\mathrm{r}+\mathrm{q}}$

(b) $\frac{r-q}{q-p}$

(c) $\frac{p-r}{p-q}$

(d) None of these

2. If $4 a^{2}+9 b^{2}+16 c^{2}=2(3 a b+6 b c+4 c a)$, where $a, b, c$ are non-zero real numbers then $a, b, c$ are in

(a) A.P.

(b) G.P.

(c) H.P.

(d) None of these

3. If $\mathrm{a}, \mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a_3}………..$ $\mathrm{a} _{2 \mathrm{n}}, \mathrm{b}$ are in $\mathrm{AP}$ and $\mathrm{a}, \mathrm{g} _{1}, \mathrm{~g} _{2}, \mathrm{~g} _{3}…….$, $\mathrm{g} _{2 \mathrm{n}}, \mathrm{b}$ are in G..P. and $\mathrm{h}$ is the single harmonic mean of $a \& b$, then $\frac{a _{1}+a _{2 n}}{g _{1} g _{2 n}}+\frac{a _{2}+a _{2 n-1}}{g _{2} g _{2 n-1}}+\ldots \ldots \ldots \ldots \ldots \ldots+\frac{a _{n}+a _{n+1}}{g _{n} g _{n+1}}$ is equal to

(a) $\frac{2 \mathrm{n}}{\mathrm{h}}$

(b) $2 \mathrm{nh}$

(c) $n h$

(d) $\frac{\mathrm{n}}{\mathrm{h}}$

4. If $0<x<\frac{\pi}{2}$, then the minimum value of

$(\sin x+\cos x+\operatorname{cosec} 2 x)^{3}$ is

(a) $27$

(b) $\frac{27}{2}$

(c) $\frac{27}{4}$

(d) None

5. Sum of certain odd consecutive positive integers is $57^{2}-13^{2}$, then the integers are

(a) $25,27,29, ……….111$

(b) $27,29 , ……….113$

(c) $29,31,33,…….. 115$

(d) None of these

6. If $x, y, z$ are three positive numbers in A.P, then the minimum value of $\frac{x+y}{2 y-x}+\frac{y+z}{2 y-z}$ is

(a) $2$

(b) $4$

(c) $\frac{1}{4}$

(d) None of these

7. If $\mathrm{n}$ arithmetic means are inserted between 50 and 200 , and $\mathrm{n}$ harmonic means are inserted between the same two numbers, then $\mathrm{a} _{2} \cdot \mathrm{h} _{\mathrm{n}-1}$ is equal to

(a) 500

(b) 5000

(c) 10,000

(d) None of these

Practice questions

1. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \ldots \ldots \ldots . \mathrm{a} _{\mathrm{n}}$ are positive real numbers whose product is a fixed number $\mathrm{c}$, then the minimum value of $a _{1}+a _{2}+\ldots \ldots . .+a _{n-1}+2 a _{n}$ is

(a) $ \mathrm{n}(2 \mathrm{c})^{1 / \mathrm{n}}$

(b) $(\mathrm{n}+1) \mathrm{c}^{1 / \mathrm{n}}$

(c) $ 2 \mathrm{nc}^{1 / \mathrm{n}}$

(d) $(\mathrm{n}+1)(2 \mathrm{c})^{1 / \mathrm{n}}$

2. If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is

(a) $\frac{1}{2 \sqrt{2}}$

(b) $\frac{1}{2 \sqrt{3}}$

(c) $\frac{1}{2}-\frac{1}{\sqrt{3}}$

(d) $\frac{1}{2}-\frac{1}{\sqrt{2}}$

3. Let $f(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}, \mathrm{a} \neq 0$ and $\Delta^{2} \mathrm{~b}^{2}-4 \mathrm{ac}$. If $\alpha+\beta, \alpha^{2}+\beta^{2} \& \alpha^{3}+\beta^{3}$ are in G.P, then

(a) $ \Delta \neq 0$

(b) $\mathrm{b} _{\Delta}=0$

(c) $\mathrm{c} _{\Delta}=0$

(d) $ \mathrm{bc} \neq 0$

4. If $\frac{\mathrm{bc}}{\mathrm{ad}}=\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}+\mathrm{d}}=3 \frac{\mathrm{b}-\mathrm{c}}{\mathrm{a}-\mathrm{d}}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in

(a) A.P

(b) G.P

(c) H.P

(d) A.G.P.

5. The $4^{\text {th }}$ term of the A.G.P. $6,8,8,………$ is

(a) $0$

(b) $12$

(c) $\frac{32}{3}$

(d) $\frac{64}{9}$

6. If $\mathrm{x}=111 \ldots . .1$ (20 digits), $\mathrm{y}=333 \ldots \ldots . .3$ (10 digits) and $\mathrm{z}=222 \ldots \ldots \ldots . .2\left(10\right.$ digits) then $\frac{\mathrm{x}-\mathrm{y}^{2}}{\mathrm{z}}=$

(a) $1$

(b) $72$

(c) $\frac{1}{2}$

(d) $3$

7. Read the passage and answer the questions that follow.

• An odd integer is the difference of two squares of integers.

• The cube of an integer is difference of two squares.

• The cube of an odd integer can be expressed as difference of two squares in two different ways.

• The difference of the cubes of two consecutive integers is difference of two squares.

(i). If $10^{3}=\mathrm{a}^{2}-\mathrm{b}^{2}$, then $\mathrm{a}-\mathrm{b}=$

(a) 5

(b) 0

(c) 10

(d) 15

(ii). If $9^{3}=a^{2}-b^{2}=c^{2}-d^{2}, a+b+c+d=$

(a) 720

(b) 750

(c) 800

(d) 810

(iii). $15^{3}-14^{3}=\mathrm{a}^{2}-\mathrm{b}^{2}, \mathrm{ab}=$

(a) 90000

(b) 95940

(c) 99550

(d) 99540

8. Match the following :-

For the given number a and b, $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ is

9. The sum of the products of the ten numbers $\pm 1, \pm 2, \pm 3, \pm 4, \pm 5$ taking two at a time is

(a) $165$

(b) $-55$

(c) $55$

(d) None of these

10. Let $a _{1}=0$ and $a _{1}, a _{2}, a _{3}, \ldots \ldots \ldots \ldots . . . a _{n}$ be real numbers such that $\left|a _{i}\right|=\left|a _{i-1}+1\right|$ for all $i$, then the A.M. of the numbers $a _{1}, a _{2}, a _{3}……………a _{n}$ has the value $A$ where

(a) $\mathrm{A}<\frac{-1}{2}$

(b) $\mathrm{A}<-1$

(c) $ \mathrm{A} \geq \frac{-1}{2}$

(d) $ \mathrm{A}=\frac{-1}{2}$