Gujarat BoardEnglish MediumSTD 10MathsArithmetic Progression3 Marks
Question
Find the sum of first $n$ even natural numbers.
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Answer
The first n even natural numbers are $2, 4, 6, 8, 10, ..., n.$
Here, $a = 2$ and $d = (4 - 2) = 2$
Sum of $n$ terms of an $AP$ is given by
$\text{S}_\text{n}=\frac{\text{n}}{2}\big[2\text{a}+(\text{n}-1)\text{d}\big]$
$=\big(\frac{\text{n}}{2}\big)\times\big[2\times2+(\text{n}-1)\times2\big]$
$=\big(\frac{\text{n}}{2}\big)\times\big[4+2\text{n}-2\big]=\big(\frac{\text{n}}{2}\big)\times(2\text{n}+2)=\text{n}(\text{n}+1)$
Hence, the required is $n(n + 1).$
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