Find the sum of all natural numbers between 1 and 1000 that are not divisible by 13.
Let S be the unknown sum and M multiples of 13 less than 1000, then S can be written in the form S=S1000−M, where S1000 is the sum of natural numbers between 1 and 1000 therefore S1000=1+2+3+...+999+1000 with a1=1 and a1000=1000
Let the nth term be
Sn=a1+an2.n....................... (i)
Sn=1+10002.1000=500500
We now proceed to solving for multiples of 13, let 13k represent all multiples of 13 and let there sum be
M=13(1)+13(2)+13(3)+...+13(k)=13.(1+2+3+...+k)............................(ii)
then
13k≤1000
k≤100013=76.923
This means that the greatest natural number k satisfying the inequality is 76.
Recall that M=13.(1+2+3+...+k)
We are going to compute the nth term of M by simply Multiplying (1) and (2)
M=13.a1+ak2.n
M=13.1+762.76=13.772.77=38038
Subtract M from Sn to obtain S
S=500500−38038=462462. The sum is therefore equal to 462462.
References
Ellina Grigoriva (2016), Methods of Solving Sequence and Series Problems, Springer international publishing AG. Pages 12-13
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