Progressions : Sum of terms of an Arithmetic Sequences

Question

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

1. 8729
2. 8700
3. 8428
4. 8278
   
   

Correct Choice is (1) and Correct Answer is 8729

Explanatory Answer

Because 4 is the remainder when we divide 200 by 7 the least number greater than 200 divisible by 7 is 203.
When we divide 400 by 7, we get a remainder of 1.

This implies that the greatest number less than 400, which is divisible by 7 is 399.

Therefore, this is an Arithmetic Progression in which the first term, t₁ = a = 203, the common difference 'd' = 7, and the last term 'l' = 399

Let 'n' be the total number of terms in this series.

The nth term of an Arithmetic Sequence is a_n = a + (n - 1)d

Then 399 = 203 + (n – 1) (7)

Simplifying we get, 7n = 399 – 203 + 7

Or 406 – 203 = 203 or n = 29.

Hence, the required sum is

Or = (29) * (301) = 8729

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