Work out the sum of all integers which are divisible by 6 lie between 1 and 1000

The sum of all integers which are divisible by 6 lie between 1 and 1000 is 83,166

Explanation:

The numbers divisible by 6 and between 1 and 1,000 are given as follows;

6, 12, ... , 996

Therefore, the numbers form an arithmetic progression with first term, a = 6 the common difference, d = 6 and the number of terms, n = 166

The sum, Sₙ, of an arithmetic progression, AP, is given as follows;

Sₙ = n/2 × (2·a + (n  - 1)·d)

Substituting the known values for the arithmetic progression formed by the numbers divisible by 6 and lie between 1 and 1,000 gives the sum of all integers which are divisible by 6 lie between 1 and 1000;

Sₙ = 166/2 × (2 × 6 + (166  - 1) × 6) = 83,166

Therefore, the sum of all integers which are divisible by 6 lie between 1 and 1000 = 83,166.

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