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/ |
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