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Sequence Sum Situation (Posted on 2021-10-06)

Define S(n) as the sum of the first n terms of an arithmetic sequence.

For some arithmetic sequence there exists positive integers m and n such that S(m) = n and S(n) = m.

What is S(m+n)?

No Solution Yet Submitted by Brian Smith

Rating: 3.0000 (1 votes)

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Solution

| Comment 1 of 4

S(m+n)= -m-n

I let the sequence have first term be a+d with constant difference d.

S(n)=na+n(n+1)d/2=m

S(m)=ma+m(m+1)d/2=n

This is just a linear system in a and d. Solving gives

a=(n^2+m^2+mn+m+n)/mn, d=-2(m+n)/mn

Writing out S(m+n) with these substitutions, the result simplifies to

S(m+n)= -m-n

This took a fair amount of algebra, so there's probably a nicer way.

Posted by Jer on 2021-10-06 13:04:22

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