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.
f0 + f2 + f4 + β¦+f2n
f0 β f1 + f2 β f3 + β¦+(-1)n fn
Mathematics Homework Solutions
#43809
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).
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.
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