Question

The fifth term of a geometric sequence is 48 and the ninth term is 768. What is the first term?

Answer 1

The first term is 3.

Explanation:

The general form of a geometric sequence with the first term `a` is `a, ar, ar^2, ar^3, ...` where `r` is the common ratio between terms.

Note that the general form for the `n`th term is `ar^(n-1)`.
Using that with our knowledge of the fifth and ninth terms, we have `ar^4 = 48` and `ar^8 = 768`

We can now eliminate `a` by dividing to obtain
`(ar^8)/(ar^4) = 768/48`
`=> r^4 = 16`

We could solve for `r` by taking the fourth root of 16, but that is not necessary for our goal of finding a.

Now that we have `r^4` we can divide once again to find `a`.
`(ar^4)/r^4 = 48/16`

`=> a = 3`