Sum of an A.P.

Concept: Finding the average or the middle term of an AP, Finding the sum of an arithmetic progression

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

Question: 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

Explanatory Answer

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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].

Find the missing information from the data given and arrive at the answer

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 correct answer.

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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