Question

. If 8 times the 8th term of an A P is equal to 12 times its 12th term, then its 20th term will be

Answer 1

Answer:

The 20th term of the A.P. is 0.

Step-by-step-explanation:

We have given that,

8 times the 8th term of AP = 12 times the 12th term of AP.

We have to find the 20th term of AP.

We know that,

$\pink{\sf\:t_n\:=\:a\:+\:(\:n\:-\:1\:)\:d}\sf\:\:\:-\:-\:[\:Formula\:]$

From the given condition,

$\sf\:8\:t_8\:=\:12\:t_{12}\\\\\\\implies\sf\:8\:\times\:[\:a\:+\:(\:8\:-\:1\:)\:d\:]\:=\:12\:\times\:[\:a\:+\:(\:12\:-\:1\:)\:d\:]\:\:\:-\:-\:[\:Using\:the\:formula\:]\\\\\\\implies\sf\:8\:\times\:(\:a\:+\:7d\:)\:=\:12\:\times\:(\:a\:+\:11d\:)\\\\\\\implies\sf\:8a\:+\:56d\:=\:12a\:+\:132d\\\\\\\implies\sf\:132d\:-\:56d\:+\:12a\:-\:8a\:=\:0\\\\\\\implies\sf\:76d\:+\:4a\:=\:0\\\\\\\implies\sf\:19d\:+\:a\:=\:0\:\:\:-\:-\:[\:Dividing\:by\:4\:]\\\\\\\implies\sf\:a\:+\:19d\:=\:0\:\:\:-\:-\:(\:1\:)$

Now,

$\sf\:t_{20}\:=\:a\:+\:(\:20\:-\:1\:)\:d\\\\\\\implies\sf\:t_{20}\:=\:a\:+\:19d\\\\\\\implies\boxed{\red{\sf\:t_{20}\:=\:0}}\sf\:\:\:-\:\:-\:-\:[\:From\:(\:1\:)\:]$

The 20th term of the A.P. is 0.

Additional Information:

1. Arithmetic Progression:

1. In a sequence, if the common difference between two consecutive terms is constant, then the sequence is called as Arithmetic Progression ( AP ).

2. $\sf\:n^{th}$ term of an AP:

The number of a term in the given AP is called as $\sf\:n^{th}$ term of an AP.

3. Formula for $\sf\:n^{th}$ term of an AP:

$\large{\boxed{\red{\sf\:t_{n}\:=\:a\:+\:(\:n\:-\:1\:)\:d}}}$

4. The sum of the first n terms of an AP:

The addition of either all the terms of a particular terms is called as sum of first n terms of AP.

5. Formula for sum of the first n terms of A. P. :

$\large{\boxed{\red{\sf\:S_{n}\:=\:\frac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}}}$