Problem Statement
If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of the first $n$ terms.
Verified Solution & Marking Scheme
Set up equations using formula Sₙ = n/2 [2a + (n - 1)d]
For $n = 7$:
$S_7 = \frac{7}{2}[2a + 6d] = 49 \implies 7(a + 3d) = 49 \implies a + 3d = 7 \quad \text{--- (1)}$
For $n = 17$:
$S_{17} = \frac{17}{2}[2a + 16d] = 289 \implies 17(a + 8d) = 289 \implies a + 8d = 17 \quad \text{--- (2)}$
Solve linear system for a and d
Subtracting (1) from (2):
$(a + 8d) - (a + 3d) = 17 - 7 \implies 5d = 10 \implies d = 2$
Substitute $d = 2$ in (1):
$a + 3(2) = 7 \implies a = 7 - 6 = 1$
Find formula for Sₙ
$S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}[2(1) + (n - 1)2] = \frac{n}{2}[2 + 2n - 2] = \frac{n}{2}[2n] = n^2$