Question

7th term of an AP is 40 . The sum of its first 13 terms is

Answer 1

ANSWER :–

sum of 13 terms = 520

EXPLANATION :–

GIVEN :–

• 7 th term of A.P. is 40.

TO FIND :–

Sum of first 13 terms = ?

SOLUTION :–

• 7th term of A.P. = 40

☞ we know that –

=> nth term of A.P. = a + (n - 1)d

• So that –

=> 7th term = a + 6d

=> a + 6d = 40 ——————eq.(1)

☞ Sum of n terms of A.P. = (n/2) [2a + (n - 1)d]

=> Sum of 13 terms = (13/2)[2×a + (13 - 1)d]

=> Sum of 13 terms = (13/2) [2a + 12d]

=> Sum = (13/2)(2)(a + 6d)

• Now using eq.(1) –

=> Sum = (13)(40)

=> Sum = 520

Answer 2

Answer:

The sum of first 13 terms of the AP is

$\boxed {\red{\sf\:S_{13}\:=\:520}}$

Step-by-step-explanation:

We have given that, the seventh term of an AP is 40 i. e. $\sf\:t_{7}\:=\:40$

We have to find the sum of first 13 terms of AP i. e. $\sf\:S_{13}$.

Now, we know that,

$\red{\sf\:t_{n}\:=\:a\:+\:(\:n\:-\:1\:)\:d}\\\\\implies\sf\:t_{7}\:=\:a\:+\:(\:7\:-\:1\:)\:d\\\\\implies\sf\:t_{7}\:=\:a\:+\:6d\\\\\implies\sf\:a\:+\:6d\:=\:40\:\:\:-\:-\:-[\:Given\:]\:-\:(\:1\:)$

Now,

$\pink{\sf\:S_{n}\:=\:\frac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}\\\\\implies\sf\:S_{13}\:=\:\frac{13}{2}\:[\:2a\:+\:(\:13\:-\:1\:)\:d\:]\\\\\implies\sf\:S_{13}\:=\:\frac{13}{2}\:[\:2a\:+\:12d\:]\\\\\implies\sf\:S_{13}\:=\:\frac{13}{2}\:\times\:2\:[\:a\:+\:6d\:]\\\\\implies\sf\:S_{13}\:=\:\frac{13}{\cancel2}\:\times\:\cancel{2}\:\times\:40\:\:\:-\:-\:-\:[\:From\:(\:1\:)\:]\\\\\implies\sf\:S_{13}\:=\:13\:\times\:40\\\\\implies\boxed {\red{\sf\:S_{13}\:=\:520}}$

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:

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

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. :

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