Consider the function f (n) = g(n, 0) g(n, x) = g(n −1, g(n −1, g(n −1, x))) if n > 0 g(0, x) = x + 1 (This is a variant of something we have done/will do in class.) Find a closed-form expression for f (n) and prove that it is correct by induction.
Consider the function f (n) = g(n, 0) g(n, x) = g(n −1, g(n −1, g(n −1, x))) if n > 0 g(0, x) = x + 1 (This is a variant of something we have done/will do in class.) Find a closed-form expression for f (n) and prove that it is correct by induction.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Consider the function
f (n) = g(n, 0)
g(n, x) = g(n −1, g(n −1, g(n −1, x))) if n > 0
g(0, x) = x + 1
(This is a variant of something we have done/will do in class.)
Find a closed-form expression for f (n) and prove that it is correct by
induction.
Further problems (for your enjoyment/practice)
•Find (and prove correct) a closed form solution for f (n) = 1 + 3 + 5 +
···+ (2n + 1).
•Prove by induction that, for any natural number n ≥0,
13+ 23+ . . . n3= (1 + 2 + ···+ n)2.
f (n) = g(n, 0)
g(n, x) = g(n −1, g(n −1, g(n −1, x))) if n > 0
g(0, x) = x + 1
(This is a variant of something we have done/will do in class.)
Find a closed-form expression for f (n) and prove that it is correct by
induction.
Further problems (for your enjoyment/practice)
•Find (and prove correct) a closed form solution for f (n) = 1 + 3 + 5 +
···+ (2n + 1).
•Prove by induction that, for any natural number n ≥0,
13+ 23+ . . . n3= (1 + 2 + ···+ n)2.
•Find (and prove correct) a closed-form solution for f (x, y), where f (0, y) =
y and
f (x, y) = 1 + f (x −1, y)
if x > 0.
•Define f (0) = 0 and f (n) = n −f (n −1) for n > 0. Find (and prove
correct) a closed-form solution for f (n).
y and
f (x, y) = 1 + f (x −1, y)
if x > 0.
•Define f (0) = 0 and f (n) = n −f (n −1) for n > 0. Find (and prove
correct) a closed-form solution for f (n).
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