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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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
Transcribed Image Text:Let ao, a1, a2, ... be a sequence of real numbers defined by the recurrence relation given
below:
do := 4
for each integer n > 1,
an = 3an-1+4
Using a Proof by Induction prove that, for all integers n > 0, the nth term of the above
sequence equals 5 – 5-".
That is, prove the following statement P(n) holds true for all integers n > 0:
P(n): а, 3D 5 — 5-т
Important: Include all relevant details in your proof. If variables appear in your proof,
clearly indicate what they represent. Clearly indicate your Induction Hypothesis and indicate
exactly when it is used in the proof of your Induction Step.
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- Solve this recurrence relation together with the initial conditions given an+2=-4an+1+5an, for n≥0, a0=2, a1=8arrow_forwardLet S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0) = S F Recursive step: If (a, b) = S, then (a, b + 1) = S, (a + 1, b + 1) = S, and (a + 2, b + 1) = S. List the elements of S produced by the first four applications of the recursive definition. Enter your answers in the form (a₁, b₁), (a2, b2),..., (an, bn), in order of increasing a, without any spaces. The first application of the recursive step adds (Click to select) ✓to S. The second application of the recursive step adds (Click to select) The third application of the recursive step adds (Click to select) The fourth application of the recursive step adds (Click to select) to S. ✓to S. ✓to S.arrow_forwardConsider the recurrence relation an = an-1 - 2an-2 with first two terms ao = 0 and a1 = 1. a. Write out the first 5 terms of the sequence defined by this recurrence relation.arrow_forward
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- 2. Let an be the number of ways to write the numbers 1, 2, 3, ..., n in order such that no number i appears immediately before i + 1 (this is a question which appeared on the exam, where an explicit formula was requested!). Explain why an will satisfy the recurrence an = (n-1) an-1+ (n − 2)an-2 for every n ≥ 2.arrow_forwardWrite a recurrence relation and initial conditions for the number Sn of sequences of Nickles, dimes and quarters that can be inserted into a vending machine to purchase a soft drink costing 5n cents. How many sequences are there for a drink costing 50 cents?arrow_forwardFind a recurrence relation for the number of sequences of yellow and purple stones of length N, where each sequence has the property that no two adjacent stones are purple. HINT: Treat sequences whose final stone is yellow separately from sequences whose final stone is purple.arrow_forward
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