Question

In an AP if Sn = n2+2n, its 18th term is
pls answer fast

Answer 1

Solution

Assuming $S_{n}$ as a sum of an arithmetic progression $\{a_{n}\}$,

by definition,

$\implies S_{n}=a_{1}+a_{2}+...+a_{n-1}+a_{n}\ \ \ \ \ (1)$

$\implies S_{n-1}=a_{1}+a_{2}+...+a_{n-1}\ \ \ \ \ (2)$

By $(1)$ and $(2)$,

$\implies S_{n}-S_{n-1}=a_{n}\ (n\geq 2)$

$\implies a_{n}=(n^{2}+2n)-\{(n-1)^{2}+2(n-1)\}\ (n\geq 2)$

$\implies a_{n}=2n+1\ (n\geq 2)$

The required value is,

$\implies a_{18}$

So the answer is,

$\implies \boxed{37}$

Learn more

$S_{n}$ having a constant term does not give an A.P when $n=1$, for example,

$\implies S_{n}=2n^{2}+3n+1$

$\implies S_{n}-S_{n-1}=(2n^{2}+3n+1)-\{2(n-1)^{2}+3(n-1)+1\}$

$\implies a_{n}=4n+1\ (n\geq 2)$

But, we see that,

$\implies a_{1}=6$

This means,

$\implies a_{1}\neq 4n+1$

Now we have two cases.

When $n=1$, by definition,

$\implies S_{1}=a_{1}$

When $n\geq 2$, by definition,

$\implies S_{n}-S_{n-1}=a_{n}$