Sequences & Series

Sequences & Series

CAT favours the two classic progressions — arithmetic (constant difference) and geometric (constant ratio) — plus a few special sums.

Arithmetic progression (AP)

$n$-th term $\;t_n=a+(n-1)d\;$ and sum

$S_n=\frac n2\big[\,2a+(n-1)d\,\big]=\frac n2\,(\text{first}+\text{last}).$

Geometric progression (GP)

$n$-th term $\;t_n=a\,r^{\,n-1}\;$ and sum

$S_n=a\,\frac{r^n-1}{r-1}\ (r\ne1);\qquad S_\infty=\frac{a}{1-r}\ \text{for}\ |r|<1.$

A worked example

Find the sum of the first 20 terms of $3,\,7,\,11,\,15,\dots$

This is an AP with $a=3$, $d=4$, $n=20$:

$S_{20}=\frac{20}{2}\big[\,2(3)+19(4)\,\big]=10\,(6+76)=10\cdot82=\mathbf{820}.$

Special sums worth memorising

$\sum_{k=1}^{n}k=\frac{n(n+1)}2,\quad \sum k^2=\frac{n(n+1)(2n+1)}6,\quad \sum k^3=\left(\frac{n(n+1)}2\right)^2.$

Common traps

• AP vs GP mix-up. Check whether terms differ by adding (AP) or multiplying (GP).
• Counting terms. From term $p$ to term $q$ inclusive there are $q-p+1$ terms — off-by-one errors are common.
• Infinite GP condition. $S_\infty$ exists only when $|r|<1$.

Checklist

• Identify AP (common difference) vs GP (common ratio)
• $S_n=\frac n2(\text{first}+\text{last})$ for any AP
• Use $\frac{a(r^n-1)}{r-1}$ for GP; $\frac{a}{1-r}$ for infinite
• Keep the $\sum k,\ \sum k^2,\ \sum k^3$ formulas ready