Question

in an AP the sum of first 10 terms is - 150 and the sum of next 10 terms is - 550 find the AP​

Answer 1

Let,

• a be the first term of the given AP

• d be the common difference of the given AP

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

Given,

• S₁₀ = -150

• n = 10

$\implies\displaystyle\sf{-150=\frac{10}{2}[2a+(10-1)d]$

$\implies\displaystyle\sf{-150=5[2a+9d]$

$\implies\displaystyle\sf{2a+9d=-30\:\:---(1)$

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$\implies\displaystyle\sf{S_{20}=-150-550$

$\implies\displaystyle\sf{S_{20}=-700$

$\implies\displaystyle\sf{-700=\frac{20}{2}[2a+(20-1)d]$

$\implies\displaystyle\sf{-700=10[2a+19d]$

$\implies\displaystyle\sf{2a+19d=-70\:\:---(2)$

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$\displaystyle\sf{(1)-(2)$

$\implies\displaystyle\sf{(2a+9d)-(2a+19d)=(-30)-(-70)$

$\implies\displaystyle\sf{2a+9d-2a-19d=-30+70$

$\implies\displaystyle\sf{-10d=40$

$\implies\displaystyle\sf{10d=-40$

$\implies\displaystyle\sf{d=-4$

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Substituting the value of d in (1),

$\implies\displaystyle\sf{2a+(9\times-4)=-30$

$\implies\displaystyle\sf{2a-36=-30$

$\implies\displaystyle\sf{2a=6$

$\implies\displaystyle\sf{a=3$

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So we get the values enough to form an A.P.

Hence,

$\boxed{\bold{\displaystyle\sf{The\: \:AP\: \:is \:\:\:\:3 \:,\: -1\: ,\: -5\: ,\: - 9\: ,\: .\: .\: .\: .}}}$