Shortcut Methods
Shortcuts and Tricks for JEE/CBSE Numericals
Arithmetic Progressions (AP)
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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: $$S_n=\frac{n}{2}(a+b)$$
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The $n$-th term, $T_n$, of an AP with first term $a$ and common difference $d$ is given by: $$T_n=a+(n-1)d$$
Geometric Progressions (GP)
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The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r$ is given by: $$S_n=\frac{a(1-r^n)}{1-r}, \quad r\neq1$$
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The $n$-th term, $T_n$, of a GP with first term $a$ and common ratio $r$ is given by: $$T_n=ar^{n-1}$$
Harmonic Progressions (HP)
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The sum of the first $n$ terms of a HP with first term $a$ and common difference $d$ is given by: $$S_n=\frac{n}{2a}[1+(n-1)d]$$
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The $n$-th term, $T_n$, of a HP with first term $a$ and common difference $d$ is given by: $$T_n=\frac{a}{n}$$
