Finding Sum of Odd Numbers

The question asks for the sum of all odd numbers between 0 and 52. These numbers form an arithmetic progression (AP).

Identifying the Arithmetic Progression

• The odd numbers strictly between 0 and 52 are: 1, 3, 5, ..., 51.
• First term ($a_1$): 1
• Last term ($a_n$): 51
• Common difference ($d$): 2 (since they are consecutive odd numbers)

Calculating the Number of Terms (n)

We use the formula for the $n^{th}$ term of an AP: $a_n = a_1 + (n-1)d$. We need to find $n$.

$51 = 1 + (n-1) \times 2$

Subtract 1 from both sides:

$51 - 1 = (n-1) \times 2$

$50 = (n-1) \times 2$

Divide both sides by 2:

$\frac{50}{2} = n-1$

$25 = n-1$

Add 1 to both sides to find $n$:

$n = 25 + 1 = 26$

Therefore, there are 26 odd numbers between 0 and 52.

Calculating the Sum of the AP

We use the formula for the sum of an AP: $S_n = \frac{n}{2}(a_1 + a_n)$.

Substitute the values $n=26$, $a_1=1$, and $a_n=51$:

$S_{26} = \frac{26}{2}(1 + 51)$

Simplify the expression:

$S_{26} = 13 \times 52$

Calculate the final sum:

$S_{26} = 676$

The sum of all odd numbers between 0 and 52 is 676.