alleryThe sum of the first $n$ terms of an AP with first term $a$ and last term $l$ is
A$S_n = n(a + l)$
B$S_n = (n/2)(a + l)$
C$S_n = n^2(a + l)/2$
D$S_n = a + l$
Answer & Solution
Correct answer: B. $S_n = (n/2)(a + l)$
1. syllabus derives the AP sum.
2. Pair up terms: 1st + last = $a + l$; 2nd + second-last = $a + l$; etc. Each pair sums to $a + l$.
3. With $n$ terms, there are $n/2$ such pairs (works whether $n$ is even or odd by symmetry).
4. Sum: $S_n = (n/2)(a + l)$.
5. Equivalently, using $l = a + (n-1)d$: $S_n = (n/2)\,[2a + (n-1)d]$.
6. Example: $1 + 2 + 3 + \ldots + 100 = (100/2)(1 + 100) = 50 \cdot 101 = 5050$ ✓ (Gauss's famous childhood calculation).
7. Other options have wrong dimensions or factors.
_Source: syllabus Mathematics, Ch 8, §8.2 (AP sum formula), p. 3–4._
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