Problem Statement
If the sum of the first $14$ terms of an AP is $1050$ and its first term is $10$, find the $20^{\text{th}}$ term of this progression.
Verified Solution & Marking Scheme
Apply the formula for S_n to find common difference d
Given: $n = 14, S_{14} = 1050, a = 10$.
$S_n = \frac{n}{2} [2a + (n - 1)d]$
$1050 = \frac{14}{2} [2(10) + (14 - 1)d] = 7 [20 + 13d]$
$\frac{1050}{7} = 20 + 13d \implies 150 = 20 + 13d$
$13d = 130 \implies d = 10$
Calculate the 20th term a_20
$a_n = a + (n - 1)d$
$a_{20} = 10 + (20 - 1)(10) = 10 + 19(10) = 10 + 190 = 200$