Is there a way to find the term number of a sum in an arithmetic series? Like if you had the sum of the first 12 terms in a series could you find the term value of term 12 using the sum?

For example in one of my previous questions :

For each series, calculate t12 and S12
a) 37 + 41 + 45 + 49 + ...

I calculated S12 to be 708. Using 708 is there a way to find t12?

3 answers

There are two common formulas for the sum of n terms of an AS

sum(n) = (n/2)(2a + (n-1)d)
and
sum(n) - (n/2)(first term + last term)

if you know S12 = 708, and we know a = 37 and n=12

708 = (12/2)(37 + last)
708 = 6(37+last)
118 = 37 + last
last = 81
but the last term would be term(12)
so term12 = 81

Thank you!!

welcome