Question

What is the sum of all positive integers that are multiples of $7$ from $200$ to $400$?

• A. $8729$
• B. $8700$
• C. $8428$
• D. $8278$

Answer 1

The correct option is A $8729$

$\frac{200}{7}$ gives the remainder = 4, which means the smallest integer between 200 and 400, divisible by 7 = 203.

Similarly$\frac{400}{7}$ gives the remainder =1, which means the smallest integer between 200 and 400, divisible by 7 = 399.

So, a series can be formed from this: 203, 210, ..., 399, which is obviously an arithmetic sequence.
Here, the first term, a = 203, common difference, d = 7

nth term of an arithmetic sequence, a(n) = a + (n - 1)d
Since 399, is the last term, so by the above formula, 399 = 203 + (n - 1)7
Solving the above equation we get, n = 29

Now, sum of the integers in an arithmetic sequence =$\left(\frac{n}{2}\right)(a+a(n\left)\right)=\left(\frac{29}{2}\right)(203+399)=8729$