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?

- 300
- 120
- 150
- 170
- 270

@ INR

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 t_{1} and d or if we can find the value of [2t_{1} + 14d].

The sum of the 4^{th} and 12^{th} term = 20.

Let t_{1} be the first term, t_{4} be the 4^{th} term, and t_{12} be the 12^{th} term of the progression.

Then t_{4} + t_{12} = 20

t_{4} can be expressed as t_{1} + 3d

Similarly, t_{12} can be expressed as t_{1} + 11d

∴ t_{4} + t_{12} = 20 can be expressed as t_{1} + 3d + t_{1} + 11d = 20

2t_{1} + 14d = 20

Sum of the first 15 terms = \\frac{15}{2}[2t_1 + 14d])

Substitute 2t_{1} + 14d = 20 in the above expression.

Sum = \\frac{15}{2} * 20) = 150.

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 = t_{8} * 15

The 8th term t_{8} = \\frac{t_4 + t_{12}}{2} = \frac{20}{2}) = 10

Hence, sum of first 15 terms = 10 * 15 = 150

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