Question

In an A.P., the sum of its first ten terms is -80 and the sum of its next ten terms is -280. Find the A.P.

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ required \ A.P. \ is \ 1, \ -1, \ -3,...}$

$\sf\orange{Given:}$

$\sf{In \ an \ A.P.,}$

$\sf{The \ sum \ of \ first \ 10 \ terms (S_{10})=-80}$

$\sf{The \ sum \ of \ next \ 10 \ terms=-280}$

$\sf\pink{To \ find:}$

$\sf{The \ A.P.}$

$\sf\green{\underline{\underline{Solution:}}}$

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

$\sf{According \ to \ the \ first \ condition}$

$\sf{-80=\frac{10}{2}[2a+(10-1)d]}$

$\sf{\therefore{5[2a+9d]=-80}}$

$\sf{\therefore{2a+9d=\frac{-80}{5}}}$

$\sf{\therefore{2a+9d=-16...(1)}}$

$\sf{According \ to \ the \ second \ condition.}$

$\sf{Sum \ of \ next \ 10 \ terms=S_{20}-S_{10}}$

$\sf{-280=\{\frac{20}{2}[2a(20-1)d]\}-(-80)}$

$\sf{\therefore{10[2a+19d]=-280+(-80)}}$

$\sf{\therefore{10[2a+19d]=-360}}$

$\sf{\therefore{2a+19d=\frac{-360}{10}}}$

$\sf{\therefore{2a+19d=-36...(2)}}$

$\sf{Subtract \ equation (2) \ from \ equation (1)}$

$\sf{2a+9d=-16}$

$\sf{-}$

$\sf{2a+19d=-36}$

_________________

$\sf{10d=-20}$

$\sf{\therefore{d=\frac{-20}{10}}}$

$\boxed{\sf{\therefore{d=-2}}}$

$\sf{Substitute \ d=-2 \ in \ equation (1)}$

$\sf{2a+9(-2)=-16}$

$\sf{\therefore{2a-18=-16}}$

$\sf{\therefore{2a=-16+18}}$

$\sf{\therefore{2a=2}}$

$\sf{\therefore{a=\frac{2}{2}}}$

$\boxed{\sf{\therefore{a=1}}}$

$\sf{The \ A.P. \ is}$

$\sf{t_{1}=a=1,}$

$\sf{t_{2}=a+d=1+(-2)=-1,}$

$\sf{t_{3}=a+2d=1+2(-2)=-3.}$

$\sf\purple{\tt{\therefore{The \ required \ A.P. \ is \ 1, \ -1, \ -3,...}}}$

Answer 2

Step-by-step explanation:

$\huge\boxed {\red {\mathbb{\underline {\underbrace {Answer}}}}}$

$<font color =purple >$

let the first term and d is the common differences of AP

= Sn=n/2{2a+(n-1)d}

now.............

=S10=10/2{2a+(10-1)d}

= -80=5(2a+9d)

= 2a+9d=-16.........(1)

again....

S20-S10=-280

=20/2{2a+19d}+80=-280

= 10{2a+19d}=-360

= 2a+19d=-36.........(2)

solve this two equation

then..

we found

d=-2

a=1

so...

AP=1,-1,-3,-5......