GMAT Quant | Arithmetic Progression Q19

GMAT Sample Questions | Sum of an AP

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

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Explanatory Answer

Step 1: Plug in available information into the AP summation formula

The sum of the first 'n' terms of an arithmetic progression = \\frac{n}{2}[2t_1 + [n - 1] \times d])
Sum of the first 15 terms of an AP = \\frac{15}{2}[2t_1 + [15 - 1] \times 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: What is the sum of the 4th and 12th term in terms of t1 and d?

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



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