(a) Find the sum of all integers between 50 and 500, which are divisible by 7. OR (b) How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3 ? Also, find their sum.

Step 1: Identify the first and last terms

For part (a), we need to find integers between 50 and 500 that are divisible by 7. The first integer greater than 50 divisible by 7 is $7 \times 8 = 56$. The last integer less than 500 divisible by 7 is $7 \times 71 = 497$. These form an arithmetic progression with a common difference of 7.

Step 2: Calculate the number of terms

To find the number of terms ($n$) in this arithmetic progression, we use the formula $n = \frac{a_n - a_1}{d} + 1$, where $a_n$ is the last term, $a_1$ is the first term, and $d$ is the common difference (which is 7). Substituting the values, we find there are 64 terms.

Step 3: Calculate the sum of the terms

Finally, we calculate the sum of these terms using the formula for the sum of an arithmetic progression: $S_n = \frac{n}{2}(a_1 + a_n)$. Plugging in the values for $n$, $a_1$, and $a_n$, we get the sum as 17696.

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