# If sum of the $3^{\text {rd }}$ and the $8^{\text {th }}$ terms of an AP is 7 and the sum of the $7^{\text {th }}$ and the $14^{\text {th }}$ terms is $-3$, find the $10^{\text {th }}$ term.

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Given:

The sum of the $3^{\text {rd }}$ and the $8^{\text {th }}$ terms of an AP is 7 and the sum of the $7^{\text {th }}$ and the $14^{\text {th }}$ terms is $-3$.

To do:

We have to find the $10^{\text {th }}$ term.

Solution:

Let $a$ be the first term and $d$ be the common difference.

This implies,

$a_{3}=a+(3-1)d$

$=a+2d$.......(i)

$a_8=a+(8-1)d$

$=a+7d$.........(ii)

$a_7=a+(7-1)d$

$=a+6d$.........(iii)

$a_{14}=a+(14-1)d$

$=a+13d$........(iv)

According to the question,

$a_3+a_8=7$

$a+2d+a+7d=7$        [From (i) and (ii)]

$2a+9d=7$.......(v)

$a_7+a_{14}=-3$

$a+6d+a+13d=-3$

$2a+19d=-3$.......(vi)

Subtracting (v) from (vi), we get,

$2a+19d-(2a+9d)=-3-7$

$10d=-10$

$d=-1$

This implies,

$2a+9(-1)=7$

$2a=7+9$

$2a=16$

$a=8$

Therefore,

$a_{10}=a+(10-1)d$

$=8+9(-1)$

$=8-9$

$=-1$

Hence, the $10^{\text {th }}$ term of the given AP is $-1$.

Updated on 10-Oct-2022 13:27:34