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

  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.

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

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