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(a_{6} + a_{12}) where a_{6} is the 6th term and a_{12} is the 12th term.

Simplifying the expression, we get 22 = a_{6} + a_{12 } ---- equation (1)

But a_{6} = a_{1} + 5d and
a_{12} = a_{1} + 11d

So, we can write equation (1) as a_{1} + 5d + a_{1} + 11d = 22

Or 2a_{1} + 16d = 22

or a_{1} + 8d = 11

From this we can determine that a_{9} = 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(a_{2} + a_{10}) where a_{2} is the 2nd term a_{10} is the 10th term of the sequence.

Simplifying the equation, we get 24 = a_{2} + a_{10}

But, a_{2} = a_{1} + d and a_{10} = a_{1} + 9d

So, 24 = a_{1} + d + a_{1} + 9d

or 24 = 2a_{1} + 10d

or 12 = a_{1} + 5d

But a_{1} + 5d = a_{6} = 12.

Hence, from statement 2 we can determine the value of a_{6}.

Statement 2 is sufficient.

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.

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

The sum of the 6th to the 12th term is 77.

Using the sum upto n terms formula we get 77 = 7/2(a

Simplifying the expression, we get 22 = a

But a

So, we can write equation (1) as a

Or 2a

or a

From this we can determine that a

Data is insufficient.

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(a

Simplifying the equation, we get 24 = a

But, a

So, 24 = a

or 24 = 2a

or 12 = a

But a

Hence, from statement 2 we can determine the value of a

Statement 2 is sufficient.

Labels: GMAT AP, GMAT Data Sufficiency, GMAT DS, GMAT Progressions, GMAT Sequence and series