GMAT Maths | Arithmetic Progressios Practice

The GMAT Math practice question given below is a sequences and series question. Concept: Sum of the first 'n' terms of an Arithmetic Progression.

Question 19: The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?

1. 300
2. 120
3. 150
4. 170
5. 270

Video Explanation

Explanatory Answer

Step 1 of solving this GMAT Arithmetic Progressions question:
Fill in available details into the summation formula

The sum of the first 'n' terms of an arithmetic progression = \\frac{n}{2}[2t_1 + [n - 1]*d] \\)
Sum of the first 15 terms of an AP = \\frac{15}{2}[2t_1 + [15 - 1]*d] \\) = \\frac{15}{2}[2t_1 + 14d] \\)
We can find the answer either if we know t₁ and d or if we can find the value of [2t₁ + 14d].

Step 2 of solving this GMAT Progressions question:
Find the missing information from the data given and arrive at the answer

The sum of the 4^th and 12^th term = 20.
Let t₁ be the first term, t₄ be the 4^th term, and t₁₂ be the 12^th term of the progression.
Then t₄ + t₁₂ = 20

t₄ can be expressed as t₁ + 3d
Similarly, t₁₂ can be expressed as t₁ + 11d
∴ t₄ + t₁₂ = 20 can be expressed as   t₁ + 3d + t₁ + 11d = 20
2t₁ + 14d = 20

Sum of the first 15 terms = \\frac{15}{2}[2t_1 + 14d] \\)
Substitute 2t₁ + 14d = 20 in the above expression.
Sum = \\frac{15}{2} * 20 \\) = 150.

Choice C is the answer.

Alternative Approach

Sum of an AP = middle term * number of terms
The middle term of an arithmetic sequence of 15 terms is the 8th term.
So, sum of first 15 terms = t₈ * 15

The 8th term t₈ = \\frac{t_4 + t_{12}}{2} = \frac{20}{2} \\) = 10
Hence, sum of first 15 terms = 10 * 15 = 150

Choice C is the correct answer.