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 $\mathrm{n}$ )

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

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

$\hspace{3.6 cm}=\ell-(\mathrm{n}-1) \mathrm{d}$

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

$\left(\mathrm{S} _{\mathrm{n}}\right.$ is a quadratic expression in $\mathrm{n}$; common difference $=\dfrac{1}{2}$ 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 $b=\dfrac{a+c}{2}$ 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 $\mathrm{d}=\dfrac{\mathrm{b}-\mathrm{a}}{\mathrm{n}+1}$.

$\mathrm{A} _{1}=\mathrm{a}+\mathrm{d}, \mathrm{A} _{2}=\mathrm{a}+2 \mathrm{~d}$ 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 $\mathrm{a} _{1} \pm \mathrm{k}, \mathrm{a} _{2} \pm \mathrm{k}, \mathrm{a} _{3} \pm \mathrm{k}, \ldots \ldots \ldots \ldots \ldots \ldots . . . \ldots$ are also in A.P.

2. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}$ are in A.P, then $a _{1} \lambda, a _{2} \lambda, a _{3} \lambda$ and $\dfrac{\mathrm{a} _{1}}{\lambda}, \dfrac{\mathrm{a} _{2}}{\lambda}, \dfrac{\mathrm{a} _{3}}{\lambda}$ are also in A.P $(\lambda \neq 0)$

3. If $\mathrm{a} _{1}, \mathrm{a} _{2}$ $\mathrm{a} _{\mathrm{n}}$ are in A.P, then $\mathrm{a} _{\mathrm{n}}, \mathrm{a} _{\mathrm{n}-}$ $\mathrm{a} _{2}, \mathrm{a} _{1}$ is also an A.P with common difference (d)

4. If $\mathrm{a} _{1}, \mathrm{a} _{2}$ $\mathrm{a} _{\mathrm{3}}$…………..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 $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}$,…………..and $\mathrm{b} _{1}, \mathrm{~b} _{2}, \mathrm{~b} _{3}$,…………..are two A.P.s then $\mathrm{a} _{1} \mathrm{~b} _{1}, \mathrm{a} _{2} \mathrm{~b} _{2}, \mathrm{a} _{3} \mathrm{~b} _{3}$…………..and $\dfrac{\mathrm{a} _{1}}{\mathrm{~b} _{1}}, \dfrac{\mathrm{a} _{2}}{\mathrm{~b} _{2}}, \dfrac{\mathrm{a} _{3}}{\mathrm{~b} _{3}}$ 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 $a-3 d, a-d, a+d, a+3 d$.

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.

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

$a _{1}+a _{n}=a _{2}+a _{n-1}=a _{3}+a _{n-2}=……….$

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

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

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

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

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

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

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

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

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

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 $r=\left(\dfrac{b}{a}\right)^{\dfrac{1}{n+1}}$

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

Properties of G.P.

1. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}$…………… are in G.P., then $\mathrm{a} _{1} \mathrm{k}, \mathrm{a} _{2} \mathrm{k}, \mathrm{a} _{3} \mathrm{k}$,………….. and $\dfrac{\mathrm{a} _{1}}{\mathrm{k}}, \dfrac{\mathrm{a} _{2}}{\mathrm{k}}, \dfrac{\mathrm{a} _{3}}{\mathrm{k}}$,…………….are also in G.P $(\mathrm{k} \neq 0)$.

2. If $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}$, are in G.P., then $\dfrac{1}{\mathrm{a} _{1}}, \dfrac{1}{\mathrm{a} _{2}}, \dfrac{1}{\mathrm{a} _{3}}$ 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 $\dfrac{1}{\mathrm{r}}$.

4. If $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots \ldots .$. and $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \ldots \ldots \ldots .$. 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 $\mathrm{b} _{1}, \mathrm{~b} _{2}, \mathrm{~b} _{3}$, 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 $\dfrac{\mathrm{a} _{1}}{\mathrm{~b} _{1}}, \dfrac{\mathrm{a} _{2}}{\mathrm{~b} _{2}}, \dfrac{\mathrm{a} _{3}}{\mathrm{~b} _{3}}$ $ \qquad $ are also in G.P.

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

$\hspace{0.7 cm}a _{1} a _{n}=a _{2} a _{n-1}=a _{3} a _{n-2}=$………….

$\hspace{0.7 cm}$ 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 $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3}, \ldots \ldots \ldots \ldots . .$. 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 $\dfrac{1}{a _{1}}, \dfrac{1}{a _{2}}, \ldots . . \dfrac{1}{a _{n}}$ are in A.P.

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

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

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

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 $b=\dfrac{2 a c}{a+c}$ is called the single H.M. between $\mathrm{a} & \mathrm{c}$. Let $\mathrm{a}$ & $\mathrm{~b}$ be two given numbers and $\mathrm{H} _{1}, \mathrm{H} _{2}, \ldots \ldots \ldots \ldots . ., \mathrm{H} _{\mathrm{n}}$ are 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

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

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

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

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

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

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

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

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

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

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

(equality holds if $\mathrm{a} _{1}=\mathrm{a} _{2}=…………\mathrm{a} _{n}$ )

Note : Also $\sqrt{\dfrac{a _{1}{ }^{2}+a _{2}{ }^{2}+\ldots \ldots .+a _{n}{ }^{2}}{n}} \geq \dfrac{a _{1}+a _{2}+\ldots \ldots . .+a _{n}}{n}$

(Root mean square inequality)

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

Note : Formation of progressions

Two consecutive terms determine the required progression. If two numbers $a & b$ are given, then

(i) a, b, 2b-a is A.P.

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

(ii) a, b, $\dfrac{\mathrm{ab}}{2 \mathrm{a}-\mathrm{b}}$ 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) $\dfrac{\mathrm{p}+\mathrm{q}}{\mathrm{r}+\mathrm{q}}$

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

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

(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 $a, a _{1}, a _{2}, a _{3},$…………$a _{2n}$, b are in AP and a, $g _{1}, g _{2}, g _{3}$,……………$\mathrm{g} _{2 \mathrm{n}}, \mathrm{b}$ are in G..P. and $\mathrm{h}$ is the single harmonic mean of $a$ & $b$, then

$\dfrac{a _{1}+a _{2 n}}{g _{1} g _{2 n}}+\dfrac{a _{2}+a _{2 n-1}}{g _{2} g _{2 n-1}}+\ldots \ldots \ldots \ldots \ldots . . .+\dfrac{a _{n}+a _{n+1}}{g _{n} g _{n+1}}$ is equal to

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

(b) 2nh

(c) nh

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

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

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

(a) 27

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

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

(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 $\dfrac{x+y}{2 y-x}+\dfrac{y+z}{2 y-z}$ is

(a) 2

(b) 4

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

(d) None of these

7. If n arithmetic means are inserted between 50 and 200, and $n$ 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

Exercise

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 $a+b+c=\dfrac{3}{2}$, then the value of $a$ is

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

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

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

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

3. Let $f(x)=a x^{2}+b x+c, a \neq 0$ and $\Delta^{2}-4 a c$. 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 $\dfrac{b c}{a d}=\dfrac{b+c}{a+d}=3 \dfrac{b-c}{a-d}$, then $a, b, c, d$ 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. $6,8,8, \ldots \ldots \ldots \ldots .$. is

(a) 0

(b) 12

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

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

6. If $x=111 \ldots . .1$ (20digits), $y=333 \ldots \ldots . . .3$ (10digits) and

$z=222 \ldots \ldots \ldots . .2\left(10\right.$ digits) then $\dfrac{x-y^{2}}{z}=$

(a) 1

(b) 72

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

(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 $b, \dfrac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ 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 $\mathrm{a} _{1}, \mathrm{a} _{2}, \mathrm{a} _{3} \ldots \ldots \ldots \ldots \ldots \mathrm{a} _{\mathrm{n}}$ has the value $\mathrm{A}$ where

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

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

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

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