In a Fibonacci sequence: 3, 4, 7, 11, 18, 29,... nth, what is the sum of n terms?For example n=3 the sum is 14; n=10 the sum is 517. what is the sum of nth term?
Let `A_n` be `n`th term of your sequence. Your sequence is recursively defined as ` `
`A_(n+1)=A_n+A_(n-1),\ n=2,3,ldots,\ \ \ A_1=3,\ A_2=4` (1)
Formula for sum of first n terms of your sequence is
`sum_(k=1)^n A_k=A_(n+2)-4` (2)
We will prove this formula by use of mathematical induction
ii) Assume that formula (2) holds for all natural numbers less or equal to `n`.
`sum_(k=1)^(n+1)A_k=sum_(k=1)^(n)A_k+A_(n+1)` now we use assumption ii)
`=A_(n+2)-4+A_(n+1)` now we use recursive relation (1)
which proves formula (2).
Of course formula (2) doesn't give you the way to calculate `A_(n+2)` without calculating all the previous terms of the sequence. For that you need to find non-recursive formula for term `A_n` which is not an easy task. If you are interested the formula for `n`th term is
`A_n=(-2)^-n (f)^(-1 -n) cdot`
`((-1)^n (f)^(2 n) (2+f) +2^n (4 (-f)^(1 + n) + 2^n (f-4) +2 (--2-f)^n (1+f)))`
If you want to know more about recursions and combinatorics in general click here.