Generalizing Common Difference

Here is a generalized formula to find the common difference. This is useful because it saves time in solving various problems.

Given an AP for which you don't know anything except that it's $m_{th}$ term is $p$ and it's $n_{th}$ term is $q$ . Then it's common difference is $\frac{p-q}{m-n}$ .

Proof: Let $a$ be the initial term of the AP and $d$ be it's common difference. Then,

$a+(m-1)d=p$

$a+(n-1)d=q$

Subtracting our second equation from our first equation

$\cancel{a}+(m-1)d-\cancel{a}-(n-1)d=p-q$

$(m\cancel{-1}-n\cancel{+1})d=p-q$

$\boxed{d=\frac{p-q}{m-n}}$

#Algebra

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$