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(A) The first $n$ odd natural numbers form an Arithmetic Progression $(AP)$: $1, 3, 5, 7, \dots, (2n-1)$.Here,the first term $a = 1$ and the common di

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$n^2$

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(A) The first $n$ odd natural numbers form an Arithmetic Progression $(AP)$: $1, 3, 5, 7, \dots, (2n-1)$.Here,the first term $a = 1$ and the common difference $d = 3 - 1 = 2$.The sum of the first $n$ terms of an $AP$ is given by the formula $S_n = \frac{n}{2} [2a + (n-1)d]$.Substituting the values $a = 1$ and $d = 2$ into the formula:$S_n = \frac{n}{2} [2(1) + (n-1)2]$$S_n = \frac{n}{2} [2 + 2n - 2]$$S_n = \frac{n}{2} [2n]$$S_n = n^2$.Therefore,the sum of the first $n$ odd natural numbers is $n^2$.

The sum of the first $n$ odd natural numbers is:

Similar Questions

Let $S_n$,$S_{2n}$,and $S_{3n}$ be the sums of $n$,$2n$,and $3n$ terms of an Arithmetic Progression $(AP)$,respectively. Prove that $S_{3n} = 3(S_{2n} - S_n)$.

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