GMAT Quant | Arithmetic Progression Q19

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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Video Explanation

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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 4^th and 12^th term in terms of t₁ and d?

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.

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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.