Question

Find the sum of all positive integers less than 200 that are divisible by 7.

Answer 1

Step-by-step explanation:

the positive integer numbers that are divisible by 7 are an arithmetic sequence by always adding 7 :

a1 = 7

a2 = a1 + 7 = 7+7 = 14

a3 = a2 + 7 = a1 + 7 + 7 = 7 + 2×7 = 21

...

an = a1 + (n-1)×7 = 7 + (n-1)×7 = n×7

the sum of an arithmetic sequence is

n/2 × (2a1 + (n - 1)×d)

with a1 being the first term (in our case 7).

d being the common difference from term to term (in our case 7).

how many terms (what is n) do we need to add ?

we need to find n, where the sequence reaches 200.

200 = n×7

n = 200/7 = 28.57142857...

so, with n = 29 we would get a number higher than 200.

so, n=28 gives us the last number divisible by 7 that is smaller than 200 (28×7 = 196).

the sum of all positive integers below 200 that are divisible by 7 is then

28/2 × (2×7 + 27×7) = 14 × 29×7 = 2,842