determine the number of terms n in each geometric series

Question

21) a_1 = -2, r = 5, S_n = -62
23) a_1 = -3, r = 4, S_n = -409
thank u! :-)

Answer 1

I'll do the first, you do the 2nd in the same way

a = -2
r = 5
Sum(n) = a(r^n - 1)/(r-1)
-62 = -2(5^n - 1)/4
124 = 5^n - 1
5^n = 125
by observation, n = 3

long way if it were not so obvious,
log 5^n = log 125
n log 5 = log 125
n = log 125/log5 = 3
take log of both sides