UPSC NDA I 2025 Mathematics Paper

Given,
The sum of the first 8 terms of a GP is 5 times the sum of its first 4 terms.
The formula for the sum of the first n terms of a GP is:
Sn=a‌

1−rn

1−r

Substitute for S8 and S4 into the equation:
‌a‌

1−r8

1−r

=5×a‌

1−r4

1−r

‌‌

1−r8

1−r4

=5
‌r8−5r4+4=0
Let x=r4, giving the quadratic equation:
x2−5x+4=0
Solving for x , we get:
x=4‌‌‌ or ‌‌‌x=1
Since x=r4, this gives:
r4=4⇒r=±√‌2
The number of possible real values of r is two: ±√‌2.