If a_ i _ i=1 ^ n where n is an even integer, is an arithmetic pr… - Vidyadip

MCQ

If $\left\{a_{i}\right\}_{i=1}^{n}$ where $n$ is an even integer, is an arithmetic progression with common difference $1$ , and $\sum \limits_{ i =1}^{ n } a _{ i }=192, \sum \limits_{ i =1}^{ n / 2} a _{2 i }=120$, then $n$ is equal to

A
$48$
✓
$96$
C
$92$
D
$104$

✓

Answer

Correct option: B.

$96$

b
$\sum \limits_{ i =1}^{ n } a _{ i }=\frac{ n }{2}\left\{2 a _{1}+( n +1)\right\}=192$

$\Rightarrow 2 a _{1}+( n -1)=\frac{384}{ n } \ldots-(1)$

$\sum \limits_{ i =1}^{ n / 2} a _{2 i }=\frac{ n }{4}\left[2 a _{1}+2+\left(\frac{ n }{2}-1\right) 2\right]=120$

$2 a _{1}+ n =\frac{480}{ n } \cdots-(2)$

From equation ($2$) and ($1$)

$1=\frac{480}{ n }-\frac{384}{ n }$

$n =480-384=96$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The sum to infinite term of the series $1 + \frac{2}{3} + \frac{6}{{{3^2}}} + \frac{{10}}{{{3^3}}} + \frac{{14}}{{{3^4}}} + \ldots \;$ is

A board has 16 squares as shown in the figure :

out of these 16 squares, two squares are chosen at random. the probality that they have no side in common is :

Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs $)=\frac{5}{9}$, is :

In a touring cricket team there are $16$ players in all including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players from these, can be chosen, so as to include three bowlers and one wicket-keeper

Consider the set of all lines $px + qy + r = 0$ such that $3p + 2q + 4r = 0$ . Which one of the following statements is true?

If both the means and the standard deviation of $50$ observations $x_1, x_2, ………, x_{50}$ are equal to $16$ , then the mean of $(x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2$ is

A straight line $L$ at a distance of $4$ units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^o$ with the line $x + y = 0$. Then an equation of the line $L$ is

If ${\tan ^{ - 1}}\frac{{1 - x}}{{1 + x}} = \frac{1}{2}{\tan ^{ - 1}}x$, then $ x =$

Let ${f_k}\,(x)\, = \frac{1}{k}({\sin ^k}\,x\, + \,{\cos ^k}\,x)$ for $k=1,2,3,...$ Then for all $x \in R,$ the value of $f_4(x) - f_6 (x)$ is equal to

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x ^3+ bx + c =0$. If $\beta \gamma=1=-\alpha$, then $b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3$ is equal to $......$.