Question

The sequence {an) is defined by ao = 1 and
An+1=2an +1 for n = 0,1,2,.... What is the value
of a4?

Answer 1

Answer: 31

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Work Shown:

Use the value of a0 to find the value of a1. The idea is you double the previous value, and then add 1.

$a_{n+1} = 2*(a_n) + 1\\\\a_{0+1} = 2*(a_0) + 1\\\\a_{1} = 2*(1) + 1\\\\a_{1} = 3$

Which is then used to find the value of a2. Follow the same process as before (double the previous value and then add 1).

$a_{n+1} = 2*(a_n) + 1\\\\a_{1+1} = 2*(a_1) + 1\\\\a_{2} = 2*(3) + 1\\\\a_{2} = 7$

This is used to find a3

$a_{n+1} = 2*(a_n) + 1\\\\a_{2+1} = 2*(a_2) + 1\\\\a_{3} = 2*(7) + 1\\\\a_{3} = 15$

Finally we can now find a4

$a_{n+1} = 2*(a_n) + 1\\\\a_{3+1} = 2*(a_3) + 1\\\\a_{4} = 2*(15) + 1\\\\a_{4} = 31$

Recursive sequences like this aren't too bad if n is small, but as n gets larger, things become more tedious. For those cases, its best to try to find a closed form equation. If not, then the next best thing is using a spreadsheet to automate the process.