Question

the sum of first 8 therms of an ap is 100 and sum of first 19 terms is 551 find ap
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Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ required \ A.P. \ is \ .\frac{-184}{11},\frac{-128}{11},\frac{-72}{11},…}$

$\sf\orange{Given:}$

$\sf{\implies{S8=100}}$

$\sf{\implies{S19=551}}$

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

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

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

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

$\sf{... formula}$

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

$\sf{100=\frac{100}{2}[2a+7d]}$

$\sf{2a+7d=100\times \ \frac{2}{100}}$

$\sf{2a+7d=2...(1)}$

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

$\sf{551=\frac{19}{2}[2a+18d]}$

$\sf{2a+18d=551\times \frac{2}{19}}$

$\sf{2a+18d=58...(2)}$

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

$\sf{2a+18d=58}$

$\sf{-}$

$\sf{2a+7d=2}$

_________________

$\sf{11d=56}$

$\sf{\implies{d=\frac{56}{11}}}$

$\sf{Substitute \ d=\frac{56}{11} \ in \ eq(1)}$

$\sf{2a+7(\frac{56}{11})=2}$

$\sf{2a+\frac{392}{11}=2}$

$\sf{Dividing \ the \ equation \ by \ 2 \ throughout}$

$\sf{a+\frac{196}{11}=1}$

$\sf{a=1-\frac{196}{11}}$

$\sf{a=\frac{11-195}{11}}$

$\sf{\implies{a=\frac{-184}{11}}}$

$\sf{t1=\frac{-184}{11}}$

$\sf{t2=t1+d=\frac{-184}{11}+\frac{56}{11}=\frac{-128}{11}}$

$\sf{t3=t2+d=\frac{-128}{11}+\frac{56}{11}=\frac{-72}{11}}$

$\sf{The \ A.P. \ is \ t1,t2,t3,...}$

$\sf{i.e.\frac{-184}{11},\frac{-128}{11},\frac{-72}{11},…}$

$\sf\purple{\tt{\therefore{The \ required \ A.P. \ is \ .\frac{-184}{11},\frac{-128}{11},\frac{-72}{11},…}}}$