Concept explainers
Consider the sequence
In this exercise you are invited to derive a closed formula for
a. Find the terms
b. In the space V of all infinite sequences of real numbers (see Example 5), consider the subset W of all sequences
c. Find all geometric sequences of the form
d. Write the Fibonacci sequence as a linear combination of geometric sequences. Use your answer to find a closed formula for
e. Explain why f,, is the integer closest to
f. Find
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Linear Algebra with Applications (2-Download)
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Intermediate Algebra
- The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for n=1,2,3,... a. Prove f1+f2+...+fn=fn+21 for all positive integers n. b. Use complete induction to prove that fn2n for all positive integers n. c. Use complete induction to prove that fn is given by the explicit formula fn=(1+5)n(15)n2n5 (This equation is known as Binet's formula, named after the 19th-century French mathematician Jacques Binet.)arrow_forwardCalculate the first eight terms of the sequences an=(n+2)!(n1)! and bn=n3+3n32n , and then make a conjecture about the relationship between these two sequences.arrow_forwardGiven the recursively defined sequence a1=1,a2=3,a3=9, and an=an13an2+9an3, use complete induction to prove that an=3n1 for all positive integers n.arrow_forward
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