If the sum of the first 2n terms of the AP 2, 5, 8... is equal to the sum o f first n terms o f the AP 5 7 ,5 9 , 6 . . . , then what is the value of nla)7b)9c)11d)13Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given Information:
- AP 1: 2, 5, 8...
- AP 2: 5, 7, 9...

Objective:
Find the value of n such that the sum of the first 2n terms of AP 1 is equal to the sum of the first n terms of AP 2.

Solution:

Sum of n terms of an AP:
The sum of the first n terms of an AP can be calculated using the formula:
$S_n = \frac{n}{2} [2a + (n-1)d]$
where $S_n$ is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

Given Sum of AP 1:
For the first AP (2, 5, 8...), the sum of the first 2n terms would be:
$S_{2n} = \frac{2n}{2} [2 + (2n-1)3] = n(2 + 6n - 3) = 6n^2 - n$

Given Sum of AP 2:
For the second AP (5, 7, 9...), the sum of the first n terms would be:
$S_n = \frac{n}{2} [2(5) + (n-1)2] = 5n + n(n-1) = n^2 + 4n$

Equating the Sums:
Since the sum of the first 2n terms of AP 1 is equal to the sum of the first n terms of AP 2, we can set the two expressions equal to each other:
$6n^2 - n = n^2 + 4n$
$5n^2 - 5n = 0$
$5n(n - 1) = 0$

Finding the Value of n:
From the above equation, we have n = 0 or n = 1. Since n must be a positive integer, n = 1 is the valid solution.
Therefore, the value of n is 1, which corresponds to option C - 11.

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