Fibonacci Sequence Question

MathStudent1999

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Find the sum of the first thirty terms of the sequence:

1, 5, 6, 11, 17, 28

if the 30th term is 2888956 and the 31st term is 4674429

I figured the pattern was tn = tn-1 + tn-2

so the first term would be a, then b, a+b, a+2b

basicly the Fibonacci Sequence. Is there away to do the question without have to list the Fibonacci Sequence up to it's 30th term?
 
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Find the sum of the first thirty terms of the sequence:

1, 5, 6, 11, 17, 18

if the 30th term is 2888956 and the 31st term is 4674429

I figured the pattern was tn = tn-1 + tn-2

so the first term would be a, then b, a+b, a+2b

basicly the Fibonacci Sequence. Is there away to do the question without have to list the Fibonacci Sequence up to it's 30th term?

Let \(\displaystyle F_n\) represent the n-th Fibonacci number. Then

\(\displaystyle \displaystyle{\sum_{i=1}^n F_i = F_{n+2} -1}\)

Look at: http://en.wikipedia.org/wiki/Fibonacci_number
 
Let \(\displaystyle F_n\) represent the n-th Fibonacci number. Then

\(\displaystyle \displaystyle{\sum_{i=1}^n F_i = F_{n+2} -1}\)

Look at: http://en.wikipedia.org/wiki/Fibonacci_number

The equation above is true only for seed numbers 1 & 1.

However it can be extended to generalized case.

a, b, a+b, a+2b, 2a+3b, 3a+5b, 5a+8b, 8a+13b.....

S = a + a(1+1+2+3+5+8....) + b(1+1+3+5+.....)

and so on.....
 
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\(\displaystyle \displaystyle{\sum_{i=1}^n F_i = F_{n+2} -5}\)

Instead of minus 1, I got minus five for this sequence.

So the sum of the first thirty terms is 4674429+2888956 - 5 which is 7563380. Is this correct?
 
\(\displaystyle \displaystyle{\sum_{i=1}^n F_i = F_{n+2} -5}\)

Instead of minus 1, I got minus five for this sequence.

So the sum of the first thirty terms is 4674429+2888956 - 5 which is 7563380. Is this correct? Correct
.
 
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