# Progressions - AP - Data Sufficiency

A data sufficiency question from Arithmetic Progression.

Question

What is the 6th term of the Arithmetic sequence?

1. The sum of the 6th to the 12th term of the sequence is 77.
2. The sum of the 2nd to the 10th term of the sequence is 108.

Correct Answer : Choice B. Statement 2 alone is sufficient; statement 1 is not sufficient.

We have to find the 6th term of an arithmetic sequence.

Let us evaluate Statement 1 first

The sum of the 6th to the 12th term is 77.
Using the sum upto n terms formula we get 77 = 7/2(a6 + a12) where a6 is the 6th term and a12 is the 12th term.
Simplifying the expression, we get 22 = a6 + a12 ---- equation (1)

But a6 = a1 + 5d and a12 = a1 + 11d

So, we can write equation (1) as a1 + 5d + a1 + 11d = 22
Or 2a1 + 16d = 22
or a1 + 8d = 11
From this we can determine that a9 = 11. However, we will not be able to find the value of the 6th term.
Data is insufficient.

Let us now evaluate Statement 2 alone

The sum of the 2nd to the 10th term of the sequence is 108
Using the sum upto n terms formula we get 108 = 9/2(a2 + a10) where a2 is the 2nd term a10 is the 10th term of the sequence.

Simplifying the equation, we get 24 = a2 + a10
But, a2 = a1 + d and a10 = a1 + 9d
So, 24 = a1 + d + a1 + 9d
or 24 = 2a1 + 10d
or 12 = a1 + 5d

But a1 + 5d = a6 = 12.

Hence, from statement 2 we can determine the value of a6.
Statement 2 is sufficient.