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Hint- The formula for the sum of an A.P is${{\text{S}}_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right)$, where n is number of terms.

We have to find out the sum of first n odd numbers.

$ \Rightarrow 1 + 3 + 5 + 7 + .........................n{\text{ terms}}{\text{.}}$

As we see that $\left( {1,3,5,7..............n} \right)$makes an A.P

Where, number of terms of the series${\text{ = n}}$

First term $\left( {{a_1}} \right) = 1$

Common difference$\left( d \right) = \left( {3 - 1} \right) = \left( {5 - 3} \right) = 2$

Now apply the formula of sum of an A.P

$

{{\text{S}}_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right) \\

\Rightarrow {{\text{S}}_n} = \dfrac{n}{2}\left( {2 \times 1 + \left( {n - 1} \right)2} \right) \\

\Rightarrow {{\text{S}}_n} = \dfrac{n}{2}\left( {2 + \left( {n - 1} \right)2} \right) = n\left( {1 + n - 1} \right) = {n^2} \\

$

Note- In such types of questions always remember the basic formulas of an A.P which is stated above, then from the given condition calculate the values of first term and common difference, then apply the formula of sum of an A.P we will get the required sum of first n odd numbers.

We have to find out the sum of first n odd numbers.

$ \Rightarrow 1 + 3 + 5 + 7 + .........................n{\text{ terms}}{\text{.}}$

As we see that $\left( {1,3,5,7..............n} \right)$makes an A.P

Where, number of terms of the series${\text{ = n}}$

First term $\left( {{a_1}} \right) = 1$

Common difference$\left( d \right) = \left( {3 - 1} \right) = \left( {5 - 3} \right) = 2$

Now apply the formula of sum of an A.P

$

{{\text{S}}_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right) \\

\Rightarrow {{\text{S}}_n} = \dfrac{n}{2}\left( {2 \times 1 + \left( {n - 1} \right)2} \right) \\

\Rightarrow {{\text{S}}_n} = \dfrac{n}{2}\left( {2 + \left( {n - 1} \right)2} \right) = n\left( {1 + n - 1} \right) = {n^2} \\

$

Note- In such types of questions always remember the basic formulas of an A.P which is stated above, then from the given condition calculate the values of first term and common difference, then apply the formula of sum of an A.P we will get the required sum of first n odd numbers.

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