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What will be the number of terms of the A.P. 63, 60, 57, ... so that their sum is 693?

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22 or 24

24 or 23

21 or 22

20 or 22

answer is C.

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Detailed Solution

Given,
The arithmetic series = 63, 60, 57,...
The sum of the series

S n = 693

The sum of the arithmetic progression formula,

S n = n 2 2 a + n − 1 d

The common difference d of the arithmetic progression,
Substituting

a 1 = 63

and

a 2 = 60

Then,

d = a 2 − a 1

⇒ a 2 − a 1 = 60 − 63 ⇒ a 2 − a 1 = − 3

The number of terms n of the arithmetic progression,
Substitute a=63 and d= -3 ,

S n

=693 in the formula,

⇒ 693 = n 2 263 + n − 1 − 3 ⇒ 1386 = n 126 − 3 n − 1

⇒ 1386 = n 126 − 3 n + 3 ⇒ 1386 = n 129 − 3 n ⇒ 1386 = 129 n − 3 n 2

⇒ 3 n 2 − 129 n + 1386 = 0 ⇒ n 2 − 43 n + 462 = 0 ⇒ n 2 − 22 n − 21 n + 462 = 0

⇒ n n − 22 − 21 n − 22 = 0 ⇒ n − 22 n − 21 = 0

n=21 and n=22.
The last term of the arithmetic progression,
When n=21,
We substitute a=63 and d= -3 and n=21 in

t n = a + n − 1 d

⇒ t 21 = 63 + 21 − 1 − 3 ⇒ t 21 = 63 − 60 ⇒ t 21 = 3

When n=22,
We substitute a=63 and d= -3 and n=22 in

t n = a + n − 1 d

⇒ t 22 = 63 + 22 − 1 − 3 ⇒ t 22 = 63 − 63 ⇒ t 22 = 0

Thus the number of terms of the A.P is 21 or 22 both because the 22nd term is zero so it will not affect the net sum.
Therefore the correct option is 3.

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