Question

3, 12, 48 are the first three terms of the geometric sequence. What is the number of factors of 4 that is in the 15th term?

Answer 1

`14`

Explanation:

The first term, `3`, does not have `4` as a factor. The second term, `12`, has `4` as one factor (it is `3` multiplied by `4`). The third term, `48`, has `4` as its factor twice (it is `12` multiplied by `4`). Therefore, the geometric sequence must be created by multiplying the preceding term by `4`. Since each term has one less factor of `4` than its term number, the `15th` term must have `14` `4`s.

Answer 2

The fifteenth term's factorization will contain 14 fours.

Explanation:

The given sequence is geometric, with the common ratio being 4 and the first term being 3.

Note that the first term has 0 factors of four. The second term has one factor of four, as it is `3xx4 = 12` The third term has 2 factors of four and so on.

Can you see a pattern here? The `n^(th)` term has (n-1) factors of four. Thus the 15th term will have 14 factors of four.

There is also another reason for this. The nth term of a G.P is `ar^(n-1).` This means that as long as a doesn't contain r in itself, the nth term will have (n-1) factors of r.