[Math] Arithmetic sequence – Find term given sum of terms, a1, and d

sequences-and-series

Working on something where I've encountered the following problem: I am given the Sn (Sum of n terms) which is 689, and I know that the first 3 terms are -7, 3, and 13. This gives me the a, -7, and the d, which is 10.

The problem is, all formulas I know and have tried re-arranging aren't working out very well.

Sn = n/2(a - l)

I don't know L.

Sn = n/2(2a+(n-1)d)
689 = n/2(2a+(n-1)d)
1378 = n(2a+(n-1)d)

I don't know the N.
I've tried long-hand, adding all the numbers and came up with the 13th term being the desired 689. However, how do I get 13 with only the given information, not doing longhand as I did? What formula do I use, and re-arrange?

Thanks in advanced.

Best Answer

Here is what you can do.

Solve as normal until you have the equation in the form ax^2+bx+c=0. This is a quadratic equation and using the quadratic formula you arrive at 13. Ignore the negative value as "n" cant be negative.

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