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.

Explanatory Answer

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.