In an A.P. the sum of $n$ terms is $5n^2−5n$. Find the $10^{th}$ term of the A.P.

Academic Mathematics NCERT Class 10

Given: In an A.P. the sum of $n$ terms is $5n^2-5n$.

To do: To find the $10^{th}$ term of the A.P.

Solution:

Sum of the $n$ terms $=5n^2-5n$

$\because S_n=\frac{n}{2}( 2a+( n-1)d)$, where $a$ is the first term.

$S_1=$ Sum of $1$ term $=5-5=0$

$\therefore a_1=0\ .......(1)$

$S_2=$ Sum of $2$ terms $=5( 2)^2-5( 2)=10$

$\therefore a_1+a_2=10......( 2)$

Solving $( 1)$ and $( 2)$

$a_2=10$

$\therefore a_2=10$

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

$\therefore a_1+d=10$

$\Rightarrow 0+d=10$

$\Rightarrow d=10$

$\therefore a_{10}=a+9d$

$=0+9(10)=90$

$\therefore a_{10}=90$

Thus, $10^{th}$ term of the A.P. is $90$.

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Updated on: 10-Oct-2022

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