Question

9) The sum of the first 51 terms of the A.P whose 2nd
term is 2 and 4th term is 8, is
​

Answer 1

Solution :-

Let :-

First term = a

Common difference = d

We know :-

$\bf a_n = a+ (n-1)d$

$\bullet \sf \ 2^{nd} \ term = 2$

$\longrightarrow \sf a+(2-1)d = 2 \\\\\longrightarrow a+d = 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...........................(1)$

$\bullet \sf \ 4^{th} \ term = 8$

$\longrightarrow \sf a+(4-1)d = 8 \\\\\longrightarrow a+3d = 8 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...........................(2)$

Equation (2) - Equation (1) :-

$\longrightarrow \sf 2d = 6\\\\\longrightarrow \bf d = 3$

Substitute the value of d in eq (1) :-

$\longrightarrow \sf a+3 = 2 \\\\\longrightarrow \bf a = -1$

We know :-

$\bf Sum \ of \ first \ n \ terms = \dfrac{n}{2} \times [ \ 2a+(n-1)d \ ]$

Sum of first 51 terms :-

$\longrightarrow \sf \dfrac{51}{2} \times [ \ 2 \times -1 + (51-1) \times 3 \ ] \\\\\longrightarrow \dfrac{51}{2} \times [ \ -2+(50 \times 3 ) \ ] \\\\\longrightarrow \dfrac{51}{2} \times [ -2+150 \ ] \\\\\longrightarrow \dfrac{51}{2} \times 148 \\\\\longrightarrow\bf 3774$

Answer 2

Given:-

• 2nd term of the A.P is 2
• 4th term of the A.P is 8

To Find:-

• The sum of the first 51 terms of the A.P

Solution:-

First term:- a

common difference:- d

We know,

$\tt a_n = a+ (n-1)d$

second term:-

$:\implies\tt a_2 = a+ (2-1)d$

$:\implies\tt a_2 = a+d=2\;\;\;\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf eq\;(1)\bigg\rgroup$

Fourth term:-

$:\implies\tt a_4 = a+ (4-1)d$

$:\implies\tt a_4 = a+3d=8\;\;\;\;\;\;\;\;\;\;\;\;\bigg\lgroup\bf eq\;(2)\bigg\rgroup$

On subtracting equation(1) from (2):-

$:\implies\tt 2d=6$

$:\implies\tt d= \dfrac{2}{6}$

Substituting the value of d in equation(1):-

$:\implies\tt a+3=2$

$:\implies\tt a=(-1)$

Sum of first 51 terms of the A.P:-

$:\implies\sf S_n = \dfrac{n}{2}\bigg(2a + (n - 1)d\bigg)$

$:\implies\sf S_n =\dfrac{51}{2}\times\bigg(2 \times -1 + (51-1) \times 3\bigg)$

$\begin{lgathered}:\implies\tt\dfrac{51}{2} \times [ \ 2 \times -1 + (51-1) \times 3 \ ] \\\\:\implies \tt\dfrac{51}{2} \times [ \ -2+(50 \times 3 ) \ ] \\\\:\implies\tt \dfrac{51}{2} \times [ -2+150 \ ] \\\\:\implies\tt \dfrac{51}{2} \times 148 \\\\:\implies\tt 3774\end{lgathered}$