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

2 Answers
Feb 13, 2016

Identify the appropriate power of 22 to find that there are 2727 terms.

Explanation:

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

a_n = a r^(n-1)an=arn1

where aa is the first term and rr is the common ratio.

In our example, a=1a=1 and r=2r=2, so the question boils down to identifying which power of 22 is 6710886467108864.

Notice that 2^10 = 1024 ~~ 1000 = 10^3210=10241000=103 and 6710886467108864 is a little over 64 * 10^664106 hence we find the correct power is:

2^26 = 2^6 * 2^10 * 2^10 = 64*1024*1024226=26210210=6410241024

So there are 2727 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).

Feb 13, 2016

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