Sum of Sequence: a, b, a+b, a+2b, 2a+3b...

Date: 01/26/2003 at 19:20:20
From: Daniel
Subject: Sum of sequence: a, b, a+b, a+2b, 2a+3b... (not Fibonacci)

Is there a formula to find the sum of the first n terms of the 
sequence a, b, a+b, a+2b, 2a+3b... ?

If I know a and b, and how many terms, is there a way to find the sum?

Question: Find the sum of the first 30 terms of the sequence: 
1, 5, 6, 11, 17, 28... if the 30th term is 2888956 and the 31st term
is 4674429.

Thanks.

Date: 01/28/2003 at 06:37:11
From: Doctor Jacques
Subject: Re: Sum of sequence: a, b, a+b, a+2b, 2a+3b... (not Fibonacci)

Hi Daniel,

Although your sequence is not the Fibonacci sequence, it looks a lot 
like it and the formula is the same:

   a[n+2] = a[n+1] + a[n]

We can also write it as:

   a[n] = a[n+2] - a[n+1]

Let us write some terms, starting at the end:

   a[29] = a[31] - a[30]
   a[28] = a[30] - a[29]
   a[27] = a[29] - a[28]
......
   a[2] = a[4] - a[3]
   a[1] = a[3] - a[2]

We notice that, on the right side, the first term of each equation 
(starting at the second) also appears with a minus sign in the 
previous equation.

This means that, if we add together all the equations, a lot of terms 
will cancel each other.

On the left-hand side, we will have a[1] + ... + a[29] - this is 
almost what we are looking for.

On the right-hand side:

  The first (a[31]) will remain there
  All the terms from a[30] down to a[3] will cancel out
  The last term (-a[2]) will remain.

To summarize, after all simplifications, we get:
   a[1] + ... + a[29] = a[31] - a[2]

As we know a[30], a[31], and, of course, a[2], you should be able to 
complete the calculation.

Please do not hesitate to write back if you don't see it.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/