# Arithmetic Progression : GMAT DS

Concept: Finding the nth term of an AP. Sum up to n terms of an AP

The GMAT DS question given below is from the topic Arithmetic Progressions. Problems on Arithmetic Progressions are quite easy if one understands that the basic concept behind AP is just an extrapolation of simple multiplication tables.

Directions for Data Sufficiency

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
3. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
5. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
##### Numbers

All numbers used are real numbers.

##### Figures

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

##### Note

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

#### 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.

Video explanation will be added soon

#### What kind of an answer will the question fetch?

We have to find the 6th term of an arithmetic sequence. The answer has to be a number.

#### When is the data sufficient?

The data is SUFFICIENT if we are able to get a UNIQUE answer for the value of 6th term of the sequence from the information in the statements.

If either the statements do not have adequate data to determine the value of 6th term or if more than one value exists based on the information in the statement, the data is NOT sufficient.

#### Statement 1: The sum of the 6th to the 12th term is 77

##### Let us evaluate Statement 1 ALONE

Using the sum upto n terms formula we get 77 = $$frac {7} {2}$$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 = a9 = 11

From this statement, we can determine that a9 = 11. However, we will not be able to find the value of the 6th term.

Statement 1 ALONE is NOT sufficient.

Eliminate choices A and D. Choices narrow down to B, C or E.

#### Statement 2: The sum of the 2nd to the 10th term of the sequence is 108

##### Let us now evaluate Statement 2 ALONE.

Using the sum upto n terms formula we get 108 = $$frac {9} {2}$$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 ALONE is SUFFICIENT.

Statement 2 ALONE is SUFFICIENT but statement 1 is NOT sufficient.