Cristofer Watson

Cristofer Watson

Answered

2022-10-25

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

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Phillip Fletcher

Phillip Fletcher

Expert

2022-10-26Added 21 answers

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

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Cory Russell

Cory Russell

Expert

2022-10-27Added 4 answers

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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