Question

if second and third terms of an A.P are 24and 28 respectively ,then find the sum of first 61 terms​

Answer 1

Answer:

8540

Step-by-step explanation:

(See in the attachment)

Answer 2

Answer:

The sum of the first 61 terms of the A.P. is 8540.

Step-by-step-explanation:

We have given that,

The second term of an A.P is 24.

The third term of an A.P is 28.

We have to find the sum of first 61 terms of the A.P.

We know that,

$\pink{\sf\:t_n\:=\:a\:+\:(\:n\:-\:1\:)\:d}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:t_2\:=\:a\:+\:(\:2\:-\:1\:)\:d\\\\\\\implies\sf\:24\:=\:a\:+\:1d\\\\\\\implies\sf\:a\:+\:d\:=\:24\:\:\:-\:-\:(\:1\:)$

Now,

$\sf\:t_3\:=\:a\:+\:(\:3\:-\:1\:)\:d\\\\\\\implies\sf\:28\:=\:a\:+\:2d\\\\\\\implies\sf\:a\:+\:2d\:=\:28\:\:\:-\:-\:(\:2\:)$

By subtracting equation ( 1 ) from equation ( 2 ), we get,

$\sf\:\cancel{a}\:+\:2d\:=\:28\:\:\:-\:-\:-\:(\:2\:)\\\sf\:-\:\\\underline{\sf\:\cancel{a}\:+\:d\:=\:24}\sf\:\:\:-\:-\:(\:1\:)\\\\\\\implies\boxed{\red{\sf\:d\:=\:4}}$

By substituting d = 4 in equation ( 1 ), we get,

$\sf\:a\:+\:d\:=\:24\:\:\:-\:-\:(\:1\:)\\\\\\\implies\sf\:a\:+\:4\:=\:24\\\\\\\implies\sf\:a\:=\:24\:-\:4\\\\\\\implies\boxed{\red{\sf\:a\:=\:20}}$

Now, we know that,

$\pink{\sf\:S_n\:=\:\dfrac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:S_{61}\:=\:\dfrac{61}{2}\:[\:2\:\times\:20\:+\:(\:61\:-\:1\:)\:\times\:4\:]\\\\\\\implies\sf\:S_{61}\:=\:\dfrac{61}{2}\:[\:40\:+\:60\:\times\:4\:]\\\\\\\implies\sf\:S_{61}\:=\:\dfrac{61}{2}\:[\:40\:+\:240\:]\\\\\\\implies\sf\:S_{61}\:=\:\dfrac{61}{\cancel2}\:\times\:\cancel{280}\\\\\\\implies\sf\:S_{61}\:=\:61\:\times\:140\\\\\\\implies\boxed{\red{\sf\:S_{61}\:=\:8540}}$

The sum of the first 61 terms of the A.P. is 8540.