# Fibonacci Sequence Proofs, Pascal's Triangle and Binomial Coefficients

Practice problem 1

Fn is the Fibonacci sequence (f0 = 0, f1 = 1, fn+1 = fn + fn-1). By considering examples, determine a formula for the following expressions, and then verify the formula.

a. f0 + f2 + f4 + ...+f2n

b. f0 - f1 + f2 - f3 + ...+(-1)n fn

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Practice problem 3

By observation, derive a formula for (n 0) + (n 1)2 + (n 2)^2 +...+(n n)2^n = the summation n where k=0 (n k)2^k. Verify your formula.

( ) are being used to express n chose zero, n chose one ...

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Practice Problem 8

Give a formula for the Fibonacci numbers using binomial coefficients (using the identity observed in Pascal's triangle).

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#### Solution Summary

Fibonacci Sequence Proofs, Pascal's Triangle and Binomial Coefficients are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.