perplexus.info :: Sequences : Sequence Sum Situation

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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Puzzle Solution

Comment 4 of 4 |

By the given conditions, we have:

S(m)=n, and S(n)=m

Denoting the first term by a, common difference by d, we have:

(m/2)*{2a+(m-1)*d}=n, .........(i)

and, (n/2)*{2a+(n--1)*d=m ......(ii)

So, (i)-(ii) gives:

a(m-n)+(1/2)*{(m^2-n^2)-(m-n) =n-m

=>2a(m-n)+{(m^2-n^2)-(m-n)}= -2(m-n)

=> (m-n){2a+(m+n-1)d} =-2(m-n)

=> 2a+(m+n-1)d= -2 ........(iii)

Consequently,

S(m+n)

= {(m+n)/2}*{2a+(m+n-1)d}

= {(m+n)/2}*(-2) ..... [from (iii)]

= -(m+n)

Posted by K Sengupta on 2022-01-14 08:07:19

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