CUET UG 2025 — General Test Quantitative Reasoning

The sum of n terms of an A.P. is given as:

$$S_n = nP + \frac{1}{2}n(n-1)Q$$Sn​=nP+21​n(n−1)Q

The standard formula for the sum of n terms of an A.P. with first term $a$a and common difference $d$d is:

$$S_n = \frac{n}{2}[2a + (n-1)d]$$Sn​=2n​[2a+(n−1)d]

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Expanding the standard formula:

$$S_n = \frac{n}{2} \times 2a + \frac{n}{2} \times (n-1)d$$Sn​=2n​×2a+2n​×(n−1)d

$$S_n = na + \frac{n(n-1)d}{2}$$Sn​=na+2n(n−1)d​

$$S_n = na + \frac{1}{2}n(n-1)d$$Sn​=na+21​n(n−1)d

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Comparing the given formula with the standard formula:

Given: $S_n = nP + \frac{1}{2}n(n-1)Q$Sn​=nP+21​n(n−1)Q

Standard: $S_n = na + \frac{1}{2}n(n-1)d$Sn​=na+21​n(n−1)d

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Since both expressions represent $S_n$Sn​ for all values of n, the corresponding coefficients must be equal.

The coefficient of $n$n gives: $P = a$P=a

The coefficient of $\frac{1}{2}n(n-1)$21​n(n−1) gives: $Q = d$Q=d

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Therefore, the common difference $= Q$=Q