If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Given: $S_7 = 49$ and $S_{17} = 289$
Using $S_n = \dfrac{n}{2}[2a + (n-1)d]$:
$$S_7 = \frac{7}{2}[2a + 6d] = 49 \implies 2a + 6d = 14 \implies a + 3d = 7 \quad \text{...(1)}$$
$$S_{17} = \frac{17}{2}[2a + 16d] = 289 \implies 2a + 16d = 34 \implies a + 8d = 17 \quad \text{...(2)}$$
Subtracting (1) from (2): $5d = 10 \implies d = 2$
From (1): $a + 6 = 7 \implies a = 1$
$$S_{20} = \frac{20}{2}[2(1) + 19(2)] = 10[2 + 38] = 10 \times 40 = \boxed{400}$$
Source: Chapter 5, Exercise 5.3 (Q9)
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Explanation
- Set up two equations using the $S_n$ formula for $n = 7$ and $n = 17$, then solve simultaneously for $a$ and $d$.
- Examiners award marks for: forming the two equations (1 mark), solving to get $a = 1, d = 2$ (1 mark), and correct $S_{20}$ (1 mark).
- Note: The textbook Q9 asks for $S_n$ in general; this question specifically asks for $S_{20}$, so substitute $n = 20$ at the end.