Name: Sum of First n Terms of an Arithmetic Progression
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: 13+13+13+13…….How do we find the sum of 50 terms of this series? Learn to find the sum of the first ‘n’ terms of an Arithmetic Progression with a real-life e...

Question

The sum of the first

n

terms of an AP whose first term is

8

and the common difference is

20

is equal to the sum of first

2 n

terms of another AP whose first term is

- 30

and the common difference is

8

. Find

n

.

Accepted Answer

Given that, first term of the first AP $(a) = 8$

and common difference of the first $A P (d) = 20$

Let the number of terms in first AP be n.

$∵$ Sum of first n terms of an $A P, S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$∴ S_{n} = \frac{n}{2} [2 \times 8 + (n - 1) 20]$

$\Rightarrow S_{n} = \frac{n}{2} (16 + 20 n - 20)$

$\Rightarrow S_{n} = \frac{n}{2} (20 n - 4)$

$∴ S_{n} = n (10 n - 2) . .. (i)$

Now, first term of the second $A P (a') = - 30$

and common term of the second $A P (d') = 8$

$∴$ Sum of first 2n terms of second $A P, S_{2 n} = \frac{2 n}{2} [2 a' + (2 n - 1) d']$

$\Rightarrow S_{2 n} = n [2 (- 30) + (2 n - 1) (8)]$

$\Rightarrow S_{2 n} = n [16 n - 68] . .. (i i)$

Now, by given condition,

Sum of first n terms of the first AP = Sum of first 2n terms of the second AP

$\Rightarrow S_{n} = S_{2 n}$ [ from Eqs. $(i)$ and $(i i)$ ]

$\Rightarrow n (10 n - 2) = n (16 n - 68)$

$\Rightarrow n [(16 n - 68) - (10 n - 2)] = 0$

$\Rightarrow n (16 n - 68 - 10 n + 2) = 0$

$\Rightarrow n (6 n - 66) = 0$

$∴ n = 11$ $[∴ n \neq 0]$

Hence, the required value of $n$ is $11$ .

The sum of the first $n$ terms of an AP whose first term is $8$ and the common difference is $20$ is equal to the sum of first $2 n$ terms of another AP whose first term is $- 30$ and the common difference is $8$ . Find $n$ .

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is -30 and the common difference is 8. Find n.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The sum of the first $n$ terms of an AP whose first term is $8$ and the common difference is $20$ is equal to the sum of first $2 n$ terms of another AP whose first term is $- 30$ and the common difference is $8$ . Find $n$ .