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If $8$ times the 8th term of an arithmetic progression is equal to $12$ times the 12th term, then find the 20th term.

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12

times to 8th term

8

times to 12th term

0

None of these

answer is C.

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Detailed Solution

Given, 8 times 8th term of AP is equal to 12 times the 12th term.
The

n^{th}

term of AP is calculated using

a_{n} = a + (n - 1) d

From given,

8 \times a_{8} = 12 \times a_{12}

\Rightarrow 8 (a + (8 - 1) d) = 12 (a + (12 - 1) d)

\Rightarrow 8 (a + 7 d) = 12 (a + 11 d)

\Rightarrow 2 (a + 7 d) = 3 (a + 11 d)

\Rightarrow 2 a + 14 d = 3 a + 33 d

\Rightarrow a = - 19 d

20th term will be,

a_{20} = a + (20 - 1) d

{\Rightarrow a}_{20} = - 19 d + 19 d

{\Rightarrow a}_{20} = 0

Hence, the 20th term is 0.
Therefore, option 3 is correct.

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