(In reply to
Solution by Jer)
The final answer (-m-n) checks out in two simple cases
Consider n = 1 and m = 2
Then the series is 2, -1, -4
S(1) = 2 and S(2) = 1, and S(3) = -3
Consider n = 1 and m = 3
Then the series is 3, 1/3, -7/3, -5
S(1) = 3 and S(3) = 1, and S(4) = -4
However, the formula for "a" looks off.
If n = 1, then a should equal S(1) = m
But substituting n = 1 gives
a = (2 + m^2 + 2m)/m
I think the initial formula for S(n) and S(m) might be the culprit.
I calculate S(n) as n terms times the average of a and (a+ (n-1)d)
= na + (n-1)nd/2
Ok, I see what happened. Jer solved as if the first n terms were
a+d, a+2d, ..., a+nd. This works, as long as one is consistent.
I formulated the series as a, a+d, ... a + (n-1)d.
So I arrived at a = (n^2 + m^2 + mn - m - n)/mn
When n = 1, this simplifies to a = m
So Jer and I have the same solution, but a different initial setup of variables
Edited on October 7, 2021, 9:48 am
re: Solution
