Maharashtra BoardEnglish MediumSTD 10MathsArithmetic Progressions3 Marks
Question
Write the sum of first n odd natural numbers.
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Answer
Let,
Odd numbers are $1, 3, 5, 7, ....., n$
Here,
First term $a = 1$
Difference $d = 3 - 1 = 2$
We know, Sum of n terms
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}[2(1)+(\text{n}-1)2]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}[2+2\text{n}-2]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}\times2\text{n}$
$\Rightarrow\ \text{S}_\text{n}=\text{n}^2$
Hence, Sum of first n odd numbers is $n^2$.
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