CUET UG 2025 — General Test Quantitative Reasoning

Let the first term of the GP be $a$a and the common ratio be $r$r, where $a > 0$a>0.

The GP is: $a, ar, ar^2, ar^3, ar^4, ...$a,ar,ar2,ar3,ar4,...

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Each term is the sum of the two preceding terms. Taking the third term:

$$ar^2 = a + ar$$ar2=a+ar

Dividing by $a$a (since $a > 0$a>0):

$$r^2 = 1 + r$$r2=1+r

$$r^2 - r - 1 = 0$$r2−r−1=0

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Using the quadratic formula:

$$r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}$$r=2(1)−(−1)±(−1)2−4(1)(−1)​​

$$r = \frac{1 \pm \sqrt{1 + 4}}{2}$$r=21±1+4​​

$$r = \frac{1 \pm \sqrt{5}}{2}$$r=21±5​​

This gives two values:

$r = \frac{1 + \sqrt{5}}{2}$r=21+5​​ or $r = \frac{1 - \sqrt{5}}{2}$r=21−5​​

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Since all terms of the GP are positive, the common ratio must be positive.

$\frac{1 - \sqrt{5}}{2} < 0$21−5​​<0 (since $\sqrt{5} > 2$5​>2)

$$\frac{1 + \sqrt{5}}{2} > 0$$21+5​​>0

Therefore, the common ratio is $r = \frac{1 + \sqrt{5}}{2}$r=21+5​​.