# Sum of Odd Numbers Formula

If a whole number is not divisible by 2 into whole numbers then those numbers are called as odd numbers. Example: 1, 3, 5, 7, and so on… are odd numbers. But, what if you want to find the sum of first n odd numbers? What formula will you use to do so?

Related Topic: Sum of Even Numbers Formula

Index

## General Formula for Odd Numbers

As we know, the odd numbers are

1, 3, 5, 7, 9, 11, …

If we observe carefully, we can find an Arithmetic Progression Sequence(AP),

Here, the AP can be made into the formula as,

The formula for odd numbers = $$2n \pm 1$$

$$\mbox{Odd Number} = \left\{ \begin{array}{ll} 2n+1 & \mbox{if n = 0, 1, 2, 3, and so on…} \\ 2n-1 & \mbox{if n = 1, 2, 3, and so on…} \\ \end{array} \right.$$

## Sum of Odd Numbers

Let us say the sequence of the n odd numbers is
=> (2k-1) + (2(k+1)-1) + …… + (2(k+n-1)-1)

From Arithmetic Progression, we know that,

$$S_n = \frac{n}{2}[2a + (n-1)d] \mbox{ OR } \frac{n}{2}[a + l]$$

Where,
$$n$$ = no. of terms
$$a$$ = First term of the A.P.
$$d$$ = Common difference
$$l$$ = Last term

And we know,
Here, a = 2k-1, d = 2, n = n & l = 2(k+n-1) – 1

So, the sum would be $$S_n = \frac{n}{2}[2(2k-1) + (n-1)2]$$

= $$S_n = \frac{n}{2}[4k -2 + 2n -2 ]$$ = $$S_n = (n)[2k -2 + n]$$

## Sum of First n Odd Numbers

Let us find the sum of first n odd numbers: ($$S_n$$),

Therefore, $$S_n = 1 + 3 + 5 + 7 + 9 + 11 + …. + (2n – 1) … (i)$$

$$S_n = \sum_{i = 1} ^{n} (2i – 1)$$

From Arithmetic Progression, we know that,

$$S_n = \frac{n}{2}[2a + (n-1)d] \mbox{ OR } \frac{n}{2}[a + l]$$

Where,
$$n$$ = no. of terms
$$a$$ = First term of the A.P.
$$d$$ = Common difference
$$l$$ = Last term

Here, $$a = 1, d = 2 \mbox{ & } l = 2n – 1$$

Therefore, $$S_n = \frac{n}{2} [a + l]$$

$$= \frac{n}{2} [1 + (2n – 1)]$$

$$= \frac{n}{2} [2n] = n^2$$

Therefore, $$S_n = n^2$$.

For,

• $$n = 1$$
• $$S_1 = 1$$ (&)
• $$S_1 = \frac{1}{2} [1 + (2\times1 – 1)] = 1$$
• $$n = 2$$
• $$S_2 = 1 + 3 = 4$$ (&)
• $$S_2 = \frac{2}{2} [1 + (2\times2 – 1)] = 4$$
• $$n = 3$$
• $$S_3 = 1 + 3 + 5 = 9$$ (&)
• $$S_3 = \frac{3}{2} [1 + (2 \times 3 – 1)] = 9$$
• $$n = 4$$
• $$S_4 = 1 + 3 + 5 + 7 = 16$$ (&)
• $$S_4 = \frac{4}{2} [1 + (2\times4 – 1)] = 16$$

And so on… .

## Solved Examples

Question 1. Find sum of first 50 odd number.

Solution. $$S_n = n^2$$

Therefore, $$S_{50} = 50^2 = 2500$$

Question 2. Find the sum of odd number between 100 & 200.

Solution. $$S_n = n^2$$

Therefore, sum of odd number between 100 & 200 = $$S_{100} – S_{50}$$.
[Because there are 50 odd number in 1 to 100 & 100 odd numbers in between 1 to 200]

$$S_{50} = 50^2 = 2500$$
$$S_{100} = 100^2 = 10000$$

Therefore, sum of odd number between 100 & 200 = 10000 – 2500 = 7500.

## FAQs

What is the sum of first n odd numbers?

Sum of ‘n’ odd numbers: $$S_n = n^2$$

What is the sum of n odd numbers?

The formula for sum between 1 and n is $$(\frac{1}{2}(n+1))^2$$.

What is the sum of the first 25 odd numbers?

$$S_{25} = n^2 = 625$$

What are Odd numbers?

Odd numbers are a set of numbers from Real Numbers where the digit cannot be divided by 2.

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