Shortcut Methods
Shortcuts and Tricks for JEE/CBSE Numericals
Arithmetic Progressions (AP)

If $a$ and $b$ are the first and last terms, and $n$ the number of terms in an AP, then the sum of the terms, $S_n$, is given by: $$S_n=\frac{n}{2}(a+b)$$

The $n$th term, $T_n$, of an AP with first term $a$ and common difference $d$ is given by: $$T_n=a+(n1)d$$
Geometric Progressions (GP)

The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r$ is given by: $$S_n=\frac{a(1r^n)}{1r}, \quad r\neq1$$

The $n$th term, $T_n$, of a GP with first term $a$ and common ratio $r$ is given by: $$T_n=ar^{n1}$$
Harmonic Progressions (HP)

The sum of the first $n$ terms of a HP with first term $a$ and common difference $d$ is given by: $$S_n=\frac{n}{2a}[1+(n1)d]$$

The $n$th term, $T_n$, of a HP with first term $a$ and common difference $d$ is given by: $$T_n=\frac{a}{n}$$