Find the sum of all integers wich are divisible by 7 and lying between 50 and 500.

Let's verify which are the integers divisible by seven,
located between 50 and 500.

Between 50 and 60 is the number
56

Between 60 and 70 is the number 63 and
70

Between 70 and 80 is 77

And
so on...

Between 490 and 500 is
497

The sum we have to calculate is
:

S = 56 + 63 + 70 + ...... +
497

We do not know the number of terms in this sum, but we
notice that the terms of the sum is the terms of an arithmetical series, whose the first
term is a1 = 56 and the common difference is d = 7.

The
numbers of term os the sum is n and the last term, an 
=497.

Let's apply the formula of the general termof an
a.s.

an = a1 + (n-1)*d

497 =
56 + (n-1)*7

We'll remove the
brackets:

497 = 56 + 7n -
7

We'll move like terms to the left side and we'll isolate
n to the right side:

497-56+7 =
7n

448 = 7n

n =
64

So, the number of terms in the sum is
64.

S64 = (a1 + a64)*64/2

S64
= (56+497)*64/2

S64 =
553*32

S64 =
17696