# The sum of first n terms of an A.P. is $5n â€“ n^2$. Find the nth term of this A.P.

Given:

The sum of first $n$ terms of an A.P. is $5n-n^{2}$.

To do:

We have to find the $n^{th}$ term of the given A.P.

Solution:

$S_{n} =5n-n^{2}$

For $n=1,\ S_{1} =5\times1-1^{2}=5-1=4$

Therefore, first term $a=4$

For $n=2,\ S_{2} =5\times 2 - 2^{2}=10-4=6$

$\therefore$ Second term of the A.P.$=S_{2} -S_{1}$

$=6-4$

$=2$

Common difference of the A.P., $d=$second term $-$ first term

$=2-4=-2$

We know that,

$a_{n}=a+(n-1)d$

$\therefore a_n=4+( n-1) \times (-2)$

$=4-2n+2$

$=6-2n$

Therefore, the $n^{th}$ term of the given A.P. is $6-2n$.

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