Sequences and Series: Tricky Questions Solved (7)

Question

Find the sum of the first $20$ terms of an arithmetic progression whose first term is $5$ and common difference is $3$.

Solution — Step by Step

Why This Works

The formula comes from a clever pairing: the first and last terms together, the second and second-to-last, etc., all give the same total $a + a_n$. There are $n/2$ such pairs, giving $S_n = (n/2)(a + a_n)$. Substituting $a_n = a + (n-1)d$ gives the formula.

Alternative Method

Use $S_n = (n/2)(a + a_n)$ directly. First compute $a_{20} = a + 19d = 5 + 57 = 62$. Then $S_{20} = 10 \times (5 + 62) = 10 \times 67 = 670$. Same answer, slightly faster if you’re given the first and last terms instead of $a$ and $d$.

Common Mistake

Students sometimes write $(n-1)$ where it should be $n$, or vice versa. Triple-check: the AP has $n$ terms total; the common difference appears $(n-1)$ times in $a_n$ because $a_n = a + (n-1)d$. The formula has $(n-1)$ inside the brackets and $n/2$ outside.