Question

in an ap the sum of first ten term is (-80) and sum of its next ten term is (-280) find the ap ?
pls with full explanation !
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Answer 1

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Answer 2

AnsweR :

A.P. = 1, -1, -3, -5,.......

$\bf{\Large{\underline{\sf{Given\::}}}}$

In an A.P. the sum of first ten term is (-80) and sum of its next ten term is (-280).

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

The A.P.

$\bf{\Large{\underline{\rm{\orange{Explanation\::}}}}}$

Formula use :

$\bf{\large{\boxed{\sf{S_{n}=\frac{n}{2} \big[2a+(n-1)d\big]}}}}$

$\Longrightarrow\sf{a_{1}+a_{2}+...............+a_{10}=-80}\\\\\\\\\Longrightarrow\sf{\cancel{\dfrac{10}{2}} \bigg[2a+(10-1)d\bigg]=-80}\\\\\\\\\Longrightarrow\sf{5\big[2a+9d\big]=-80}\\\\\\\\\Longrightarrow\sf{2a+9d\:=\:\cancel{\dfrac{-80}{5} }}\\\\\\\\\Longrightarrow\sf{\pink{2a+9d=-16.......................(1)}}$

Or

$\Longrightarrow\sf{a_{11}+a_{12}+..............+a_{20}=-280}\\\\\\\\\Longrightarrow\sf{S_{20}-S_{10}=-280}\\\\\\\\\Longrightarrow\sf{\cancel{\dfrac{20}{2}} \bigg[2a+(20-1)d\bigg]-(-80)=-280}\\\\\\\\\Longrightarrow\sf{10\big[2a+19d\big]=-280-80}\\\\\\\\\Longrightarrow\sf{10[2a+19d]=-360}}\\\\\\\\\Longrightarrow\sf{2a+19d=\cancel{\dfrac{-360}{10} }}\\\\\\\\\Longrightarrow\sf{\pink{2a+19d=-36......................(2)}}}$

$\bigstar$ Subtracting equation (2) from equation (1), we get;

$\longmapsto\sf{\cancel{2a}+9d\cancel{-2a}-19d=-16+36}\\\\\\\longmapsto\sf{9d-19d\:=\:20}\\\\\\\longmapsto\sf{-10d\:=\:20}\\\\\\\longmapsto\sf{d\:=\:\cancel{-\dfrac{20}{10} }}\\\\\\\longmapsto\sf{\pink{d\:=\:-2}}$

$\bigstar$ Putting the value of d in equation (1), we get;

$\longmapsto\sf{2a+9(-2)\:=\:-16}\\\\\\\longmapsto\sf{2a+(-18)\:=\:-16}\\\\\\\longmapsto\sf{2a-18\:=\:-16}\\\\\\\longmapsto\sf{2a\:=\:-16+18}\\\\\\\longmapsto\sf{2a\:=\:2}\\\\\\\longmapsto\sf{a\:=\:\cancel{\dfrac{2}{2} }}\\\\\\\longmapsto\sf{\pink{a\:=\:1}}$

Thus,

The A.P. is 1, -1, -3, -5,...............