Cristofer Watson

2022-10-25

Find the number of terms in the following geometric series: 1 + 2 + 4 + ... + 67108864

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Phillip Fletcher

Expert

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 ${2}^{10}=1024\approx 1000={10}^{3}$ and 67108864 is a little over $64\cdot {10}^{6}$ hence we find the correct power is:
${2}^{26}={2}^{6}\cdot {2}^{10}\cdot {2}^{10}=64\cdot 1024\cdot 1024$
So there are 27 terms: ${2}^{0},{2}^{1},{2}^{2},...,{2}^{26}$

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Cory Russell

Expert

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).

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