Question

How many terms of the A.P. 27, 24, 21, ..... must be taken so that their sum is 105? Which term of the A.P. is zero?

Answer 1

Given that,

AP 27, 24, 21, .....

For the given AP,

First term, a = 27

Common difference, d = 24 − 27 = −3

Sum, S = 105      ...[Given]

Let n number of terms be taken for sum to be 105

So, we have,

105 = `n/2[2 xx 27 + (n − 1)(−3)]`

210 = n[54 − 3n + 3]

210 = n(57 − 3n)

210 = 57n − 3n²

3n² − 57n + 210 = 0

n² − 19n + 70 = 0

⇒ n² − 14n − 5n + 70 = 0

⇒ n(n − 14) − 5(n − 14) = 0

⇒ (n − 14)(n − 5) = 0

⇒ n = 5, 14

So, 5 and 14 terms of the given AP must be taken to get sum as 105.

Now,

Let p^th term of the AP be zero

⇒ a_p = 0

⇒ a + (p − 1)d = 0

⇒ 27 + (p − 1)(−3) = 0

⇒ 27 − 3p + 3 = 0

⇒ 30 = 3p

⇒ p = 10

Hence, 10^th term of the given AP is zero.