Find the number of terms:

How many terms of the arithmetic progression 63, 60, 57, … be taken so that their sum is 693?

Here first term a = 63 and common difference d = 60 – 63 = -3.
Also, S_n = 693
Since the sum of n terms is
S_{n =}

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

693 × 2 = n[ 126 + (n - 1)(-3)]
1386 = 126n - 3n(n - 1)
1386 = 126n - 3n² + 3n
1386 = -3n² + 129n
3n² - 129n + 1386 = 0
n² - 43n + 462 = 0
⇒ n = 21, n = 22
∴ n = 21 or n = 22

Hence, Either 21 or 22 terms of the arithmetic progression must be taken to get sum as 693.