Re: number sequence

It is a sequence with the following definition
a(1) = 2
a(n+1) = a(n) + 4^n

a(n+1)
= a(n) + 4^n
=a(n-1) + 4^(n-1) + 4^n
=a(n-2) + 4^(n-2) + 4^(n-1) + 4^n
= ...
= 2 + sum(i=1 to n-1) 4^i

a(1) = 2
a(2) = 2 + 4^1 = 6
a(3) = 2 + 4^1 + 4^2 = 2 + 4 +16 = 22
a(4) = 2 + 4^1 + 4^2 + 4^3 = 2 + 4 + 16 + 64 = 86
a(5) = 2 + 4^1 + 4^2 + 4^3 + 4^4 = 2 + 4 + 16 + 64 + 256 = 342

The equation can furthur simplified using the sum of geometric series
method
If Sn = a + ar + ar^2 + ... + ar^(n-1)
then Sn = a * (r^n - 1) / (r - 1)

2 + sum(i=1 to n-1) 4^i
= 2 + (4^1 + 4^2 + ... 4^(n-1))
= 2 + 4 * (4^(n-1) - 1) / (4 - 1)
= 2 + (4^n - 4)/3

a(1) = 2 + (4^1 - 4)/3 = 2
a(2) = 2 + (4^2 - 4)/3 = 2 + 4 = 6
a(3) = 2 + (4^3 - 4)/3 = 2 + 20 = 22
a(4) = 2 + (4^4 - 4)/3 = 2 + 84 = 86
a(5) = 2 + (4^5 - 4)/3 = 2 + 340 = 342

I hope that "2 + (4^n - 4)/3" is the equation you are looking for.

.

Relevant Pages

  • Re: Need help
    ... What is the sum of: ... > A Google search came up with over 200,000 hits. ... I would guess 1/2 or 0.50....., but check using the geometric series ... Prev by Date: ...
    (sci.math)
  • Re: a cos question
    ... sum cosis a geometric series. ... Differentiate. ... Prev by Date: ...
    (sci.math)
  • Re: Idle series question
    ... > sequence converges to a nonzero constant, ... example, sum ^k/sqrtconverges, but sum ... Prev by Date: ...
    (sci.math)
  • Re: closed form for convolution of a^n with Fibonacci numbers
    ... though...I know the closed form ... >and the formula for the sum of a geometric series, ... Now you wanted the convolution of this sequence and the sequence ... Those sums are finite geometric series, ...
    (sci.math)
  • Re: Simple proof
    ... the sequence will look like this. ... If there is another way to write n as the sum of two numbers then ... algebra. ... Prev by Date: ...
    (sci.math)