Question

Let g be a geometric sequence of integers with first term 1, and let G be the series associated with g. Let the natural number n be such that gn=78125 and Gn=97656. What is the value of n? GEOMETRIC SEQUENCES

Answer 1

n = 8

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

___

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.

_____

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