If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
Given: $S_6 = 36$, $S_{16} = 256$
Using formula $S_n = \dfrac{n}{2}[2a + (n-1)d]$
From $S_6 = 36$:
$$\frac{6}{2}[2a + 5d] = 36$$
$$3[2a + 5d] = 36$$
$$2a + 5d = 12 \quad \text{...(i)}$$
From $S_{16} = 256$:
$$\frac{16}{2}[2a + 15d] = 256$$
$$8[2a + 15d] = 256$$
$$2a + 15d = 32 \quad \text{...(ii)}$$
Subtracting (i) from (ii):
$$10d = 20 \implies d = 2$$
Substituting in (i):
$$2a + 5(2) = 12 \implies 2a = 2 \implies a = 1$$
Sum of first 10 terms:
$$S_{10} = \frac{10}{2}[2(1) + 9(2)] = 5[2 + 18] = 5 \times 20 = \boxed{100}$$
Source: Chapter 5, Exercise 5.3
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Explanation
- Examiners award marks at each step: setting up both equations (1 mark each), solving for $d$ and $a$ (1 mark each), and finding $S_{10}$ (1 mark) — total 5 marks.
- Always write the formula first, substitute clearly, and show subtraction of equations explicitly — these are scoring steps.
- This question is structurally identical to Q.9 of Exercise 5.3 (sum of $n$ terms); practise that pattern.