Question

If the sum of $7$ terms of an A.P is $49$ and that of $17$ terms is $289$, find the sum of n terms.
A.$n$
B.${n^2}$
C.${n^3}$
D.${n^4}$

Answer 1

Hint: Use the formula of A.P. for $n = 7$ and $n = 17$ to get the value of the first term and the common difference and calculate the sum of n terms

Complete step-by-step answer:

We know that the formula for the sum of n terms of Arithmetic Progression as
$S_n^{} = {\text{ }}\dfrac{n}{2}[2a + (n - 1)d]$
In this formula $a =$first term of A.P. , $n =$number of terms in A.P. , $d =$common difference of A.P.,
$S_n^{} =$sum of $n$terms
Now we should substitute the value of n as $7$ and sum as $49$ in the above formula
and get

$S_7^{} = \dfrac{7}{2}[2a + (7 - 1)d] = 49$
Now on solving this equation we get as
$\dfrac{{14 - 6d}}{2} = a$
Now forming the equation we get
$\Rightarrow a = 7 - 3d..........\left( 1 \right)$
Now on substituting the value of n as $17$and sum as $289$in the formula of Arithmetic Progression we
get
$S_{17}^{} = \dfrac{{17}}{2}[2a + (17 - 1)d] = 289$
Now we solve this equation and get

$\dfrac{{34 - 16d}}{2} = a$
\Rightarrow a = 17 - 8d........(2) \\\

 Now equating equation 1 and 2 we get   
 d=2 and then putting in equation 1 we get   
 a=1  
 now again substituting the value of a and d in formula of sum of A.P. and we get   
 $$  

S_n^{} = {\text{ }}\dfrac{n}{2}[2a + (n - 1)d] \\
{\text{ = }}\dfrac{n}{2}[2(1) + (n - 1)2] \\
{\text{ = }}\dfrac{n}{2}(2n) \\
\Rightarrow S_n^{} = {n^2} \\

$$Hence answer is option B Note: while solving questions of A.P. we should use the formula of the general sum and n term and then calculate the value of first term and common difference and then calculate the sum and get the result.$$