NCERT · Class 10 · Mathematics · Arithmetic ProgressionsFind the sum of all odd integers between 0 and 50.

Step 1: Identify the sequence of odd integers between 0 and 50.\nThe odd integers between 0 and 50 are: 1, 3, 5, 7, ..., 49.\nThis forms an Arithmetic Progression where:\nFirst term, $a = 1$\nCommon difference, $d = 3 - 1 = 2$\nLast term, $a_n = 49$ \nStep 2: Find the total number of terms ($n$).\nUsing the $n$-th term formula: $a_n = a + (n - 1)d$ $49 = 1 + (n - 1) \times 2$ $49 - 1 = 2(n - 1)$ $48 = 2(n - 1)$ $n - 1 = \frac{48}{2} = 24$ $n = 24 + 1 = 25$ \nStep 3: Calculate the sum of these 25 terms using the sum formula. $S_n = \frac{n}{2}(a + l)$ $S_{25} = \frac{25}{2}(1 + 49)$ $S_{25} = \frac{25}{2}(50)$ $S_{25} = 25 \times 25$ $S_{25} = 625$ \nConclusion: The sum of all odd integers between 0 and 50 is 625.