If eight times the $8^{th}$ term of an arithmetic progression is equal to seventeen times the $7^{th}$ term of the progression, find the $15^{th}$ term of the progression.

Given: If eight times the $8^{th}$ term of an A.P. is equal to seven times of its $7^{th}$ term.

To do: To find the $15^{th}$ term.

Solution:

As given, $8( a_8) = 7( a_7)$

$\Rightarrow 8( a + 7d) = 7 ( a + 6d)$

$\Rightarrow 8a + 56d = 7a + 42d$

$\Rightarrow 8a - 7a = 42d - 56d$

$\Rightarrow a = -14d$

The value of $15^{th}$ term $a_{15}=a + 14d$

$\Rightarrow a_{15}= -14d + 19d$

$\Rightarrow a_{15}= 0$

Therefore, $a_{15}$ is Equal to $0$.

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