Question

If an Ap whose first term is -27,the tenth term is equal to the sum of the first nine terms calculate the common difference.

Answer 1

$\sf\large\underline{Given:}$

• $\sf{The\:1st\:term\:of\:AP=-27}$
• $\sf{The\:10th\: term=The\:sum\:of\:1st\:9th\:term}$

$\sf\large\underline{To\: Find:}$

• $\sf{The\:common\: difference\:of\:AP}$

$\sf\large\underline{Let:}$

• $\sf{The\:common\: difference\:of\:AP=d}$

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

$\sf\large\underline{Formula\: used:}$

$\rm{\implies T_n=a+(n-1)d}$

$\rm{\implies S_n=\dfrac{n}{2}[2a+(n-1)d]}$

• [As per above information:-]

$\rm{\implies T_{10}=-27+(10-1)d}$

$\rm{\implies T_{10}=-27+9d}$

$\rm{\implies S_9=\dfrac{9}{2}[2*(-27)+(9-1)d]}$

$\rm{\implies S_9=\dfrac{9}{2}[-54+8d]}$

• [Now, solving T_10=S_9 here]

$\rm\purple{\implies T_{10}=S_9}$

$\rm{\implies -27+9d=\dfrac{9}{2}[-54+8d]}$

$\rm{\implies -27+9d=\dfrac{9*(-54)}{2}+\dfrac{9*8d}{2}}$

$\rm{\implies -27+9d=\dfrac{-486}{2}+\dfrac{72d}{2}}$

$\rm{\implies -27+\dfrac{486}{2}=\dfrac{72d}{2}-9d}$

$\rm{\implies -27+243=\dfrac{72d-18d}{2}}$

$\rm{\implies 216=\dfrac{54d}{2}}$

$\rm{\implies 54d=216*2}$

$\rm{\implies \cancel{54}d=\cancel{432}}$

$\rm\red{\implies d=8}$

Hence,

$\rm\orange{\implies The\: common\:difference\:of\:AP=8}$

Answer 2

If an Ap whose first term is -27,the tenth term is equal to the sum of the first nine terms calculate the common difference.

Given

• First termA=-27
• T-10=S-9

Let:-

• Common difference be d

by solving upon we get

• the common difference is 8

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