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

Question

Phillip Fletcher · Accepted Answer

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

Cory Russell · Accepted Answer

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 &#039;fun&#039; ones are             2      25        =    33554432   and             2      29        =    536870912   (which contains all the digits except 4).

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

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