Problem Statement
In an Arithmetic Progression (AP), the sum of the first 7 terms is 49 and that of the first 17 terms is 289. Find the sum of the first $n$ terms.
Verified Solution & Marking Scheme
Formulate Equations
$S_7 = \frac{7}{2}(2a + 6d) = 49 \implies a + 3d = 7$ (1).
$S_{17} = \frac{17}{2}(2a + 16d) = 289 \implies a + 8d = 17$ (2).
Solve for a and d
Subtracting (1) from (2): $5d = 10 \implies d = 2$. From (1): $a + 6 = 7 \implies a = 1$.
Calculate Sn
$S_n = \frac{n}{2}[2(1) + (n - 1)2] = \frac{n}{2}[2n] = n^2$