# what is the sum of all natural numbers between 200 and 500 which are divisible by 7? We are asked to find the sum of all natural numbers between 200 and 500 that are divisible by 7:

Note that the list of such numbers forms an arithmetic sequence; the first term is 203, the last term is 497, and the common difference is 7.

The sum of a finite arithmetic series can be found using the formula `S_(n)=(n(a_1+a_n))/2 ` where n is the number of terms, a1 is the first term of the series and an is the nth term of the series.

We know the first and last term, so we need the number of terms. Since the first term is 203 with a common difference of 7, we have ` 497=203+(n-1)7 `  so we find that the number of terms is n=43.

So the sum is `S=(43(203+497))/2=15050 `

----------------------------------------------------------------------------

An alternative is to use summation notation and the associated properties:

`sum_(i=29)^(71)7i=7sum_(i=29)^(71)i=7[((71)(72))/2- ((28)(29))/2]=15050 `

Approved by eNotes Editorial Team