Question

the 8th term of an ap whose sum upto n terms is 2n^2+n is given by

Answer 1

Given:

$S_n$ $= 2n^2 + n$

To find:

The 8th term of A.P. i.e., $a_8$

Solution:

We know the formula for the nth tern of an A.P. is given as,

$\boxed{\bold{a_n\:=\:S_n\:-\:S__(n-1)}}$

Since we have to find the 8th term of an A.P. so, we will substitute n = 8 in $S_n$ & $S__(n-1)$.

∴ $S_8$ = $2(8)^2 + 8$ ....... [∵ $S_n = 2n^2 + n$ (given) ]

⇒ $S_8$ = $(2\times64) + 8$

⇒ $S_8$ = $128 + 8$

⇒ $S_8$ = $136$ ....... (i)  

and

∴ $S__(8-1)$ = $2(8-1)^2 + (8-1)$ ....... [∵ replacing n by n-1 in → $S_n = 2n^2 + n$ ]

⇒ $S_7$ = $2(7)^2 + 7$

⇒ $S_7$ = $(2\times49) + 7$

⇒ $S_7$ = $98 + 7$

⇒ $S_7$ = $105$ ....... (ii)  

Now, substituting n = 8 in the formula of the nth term of an A.P., we get

$a_8 = S_8 - S__(8-1)$

⇒ $a_8 = S_8 - S_7$

substituting the values from (i) & (ii), we get

⇒ $a_8 = 136 - 105$

⇒ $\bold{a_8 = 31}$

Thus, the 8th term of an A.P. whose sum up to n terms is 2n² + n is 31.

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