Question

The sum of the first 10 terms of an AP is the same as the sum of its first 5 terms, then the sum of its 15th term will be _______

(a) 1

(b) 0

(c) 2

(d) 3

The question was posed to me in semester exam.

My question is based upon Sum of N Terms of Arithmetic Progression in section Arithmetic Progressions of Mathematics – Class 10

Answer 1

Right option is (b) 0

For explanation I would say: The sum of the first 10 terms of an AP is the same as the sum of its first 5 terms

S10 = S5

\(\frac {10}{2}\) (2a + (10 – 1)d) = \(\frac {5}{2}\) (2a + (5 – 1)d)

5(2a + 9d) = \(\frac {5}{2}\) (2a + 4d)

5(2a + 9d) = 5(a + 2d)

(2a + 9d) = (a + 2d)

2a – a = 2d – 9d

a =  – 7d

S15 = \(\frac {n}{2}\) (2a + (n – 1)d)

= \(\frac {15}{2}\) (2a + (15 – 1)d)

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

= 15(a + 7d)

= 15(-7d + 7d) = 0