Question

How do you find the number of terms in the following geometric series: 1 + 2 + 4 + ... + 67108864?

Answer 1

Identify the appropriate power of `2` to find that there are `27` terms.

Explanation:

The general form of a term of a geometric sequence or series is:

`a_n = a r^(n-1)`

where `a` is the first term and `r` is the common ratio.

In our example, `a=1` and `r=2`, so the question boils down to identifying which power of `2` is `67108864`.

Notice that `2^10 = 1024 ~~ 1000 = 10^3` and `67108864` is a little over `64 * 10^6` hence we find the correct power is:

`2^26 = 2^6 * 2^10 * 2^10 = 64*1024*1024`

So there are `27` terms: `2^0, 2^1, 2^2,...,2^26`

Footnote

As a child, I used to like to write powers of `2` on a blackboard, starting with `1` and doubling it repeatedly.

In later life I found it useful to memorise powers of `2` up to about `2^32 = 4294967296`.

A couple of 'fun' ones are `2^25 = 33554432` and `2^29 = 536870912` (which contains all the digits except `4`).

Answer 2

`n=27`

Explanation:

It was calculated that the `n^(th)` term in a geometric sequence or series is
`a_n=ar^(n-1)`
where `a` is the first term and `r` is the common ratio.
In the given question, `a=1 and r=2`. Therefore

(To take it further from where @George C. left in his solution)

`67108864=1*2^(n-1)`
`=>67108864=2^(n-1)`

To find `n`, taking log of both sides
`log 67108864=log 2^(n-1)`
`=> log 67108864=(n-1)log 2^`
`=> (n-1)=log 67108864/log 2`
`=> (n-1)=7.82678/0.30103`
`=> (n-1)=26`
or `n=27`