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The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000?

 May 17, 2020
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The first term is 1 and each successive term is the sum of all the previous terms:

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

 

Although not true for the first term, each of the following terms has a value of 2n - 2

where n is the number of the term.

We want       2n - 2  >=  5000

              log(2n - 2)  >=  log(5000)

          (n - 2)log(2)  >=  log(5000)

                      n - 2  >=  log(5000)/log(2)

                      n - 2  >= 12.29

                            n  >=  14.29

 

So:  n  =  15

 May 17, 2020

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