Question

The sum of the first 12 terms of an AP is 15 and the sum of its first 15 terms is 12, then the sum of its first 27 terms will be __________

(a) -24

(b) 24

(c) 27

(d) -27

The question was posed to me in quiz.

Question is taken from Sum of N Terms of Arithmetic Progression in division Arithmetic Progressions of Mathematics – Class 10

Answer 1

Right option is (d) -27

Easy explanation: The sum of the first 12 terms of an AP is 15

S12 = 15

\(\frac {n}{2}\) (2a + (n – 1)d) = 15

\(\frac {12}{2}\) (2a + (12 – 1)d) = 15

6(2a + 11d) = 15

12a + 66d = 15

4a + 22d = 5     (1)

The sum of its first 15 terms is 12

S15 = 12

\(\frac {n}{2}\) (2a + (n – 1)d) = 12

\(\frac {15}{2}\) (2a + (15 – 1)d) = 12

\(\frac {15}{2}\) (2a + 14d) = 12

15(a + 7d) = 12

15a + 105d = 12

5a + 35d = 4     (2)

On subtracting (2) from (1)

4a + 22d – (5a + 35d) = 5 – 4

-a – 13d = 1

a + 13d = -1     (3)

Now, S27 = \(\frac {n}{2}\) (2a + (n – 1)d) = \(\frac {27}{2}\) (2a + (27 – 1)d) = \(\frac {27}{2}\) (2a + 26d) = 27(a + 13d) = -27  from (3).