Answer:
 n = 8
Step-by-step explanation:
If the common ratio is r, then the term gn is ...
 gn = 1·r^(n-1)
and the sum Gn is ...
 Gn = 1·(r^n -1)/(r -1)
Using the given values for gn and Gn, we can write ...
 r^n = r·gn
so ...
 Gn = 97656 = (78125r -1)/(r -1)
Multiplying by r-1 gives
 97656r -97656 = 78125r -1
 19531r = 97655 . . . . . add 97656-78125r
 r = 5 . . . . . . . . . . . . . . divide by 19531
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Now we can find the value of n. Let's use gn.
 78125 = 5^(n-1) = 5^7
Matching exponents, we find ...
 n -1 = 7
 n = 8 . . . . . . add 1
The value of n is 8.
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Comment on power of 5
You can use logarithms to find what power of 5 gets you 78125. Taking the log of the above equation, you get
 log(78125) = (n-1)log(5)
 log(78125)/log(5) = n-1
 7 = n -1 . . . . . . evaluate the ratio of the logs
