8. Suppose that họ, h1, h2, . . . is a sequence défihed as follows: ho = 1, h = 2, h2 = 3, h = hx-1+h-2+hk-3 for each integer k > 3. %3D a. Prove that h, < 3" for every integer n 0.
8. Suppose that họ, h1, h2, . . . is a sequence défihed as follows: ho = 1, h = 2, h2 = 3, h = hx-1+h-2+hk-3 for each integer k > 3. %3D a. Prove that h, < 3" for every integer n 0.
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
Please help me solve this, using the way in the example problem.
Transcribed Image Text:8. Suppose that ho, h¡, h2, . . is a sequence defined
as follows:
ho = 1, h = 2, h2 = 3,
h = hg-1+hx-2+ hx-3 for each integer k 3.
%3D
a. Prove that h, < 3" for every integer n 0.
Transcribed Image Text:Let P(n) be a property that is defined for integers n, and let a and b be fixed integers with a b.
Suppose the following two statements are true:
1. P(a), P(a + 1),.., and P(b) are all true. (basis step)
2. For every integer & b, if PO is true for each integer i from a through k, then P(k +1) is true.
(inductive step)
Then the statement
for every integer n2a, P(n)
is true. (The supposition that P(i) is true for each integer i from a through k is called the inductive
hypothesis. Another way to state the inductive liypothesis is to say that Pla), P(a -1), P() are
all true.)
4. Suppose that di , da, da, .. is a sequence defined as follows:
10
10
de=d1 d for every integer k > 3.
Prove that 0 < d, 1 for each integer n 2 1.
ca
Proof Chy shong mmdnchion).
Assume PlD is tue fo sisk. HTS PLEA0 is tue
Let kE 2zta Kz2. Assume 0<diel, NTS O< detiel
dk= dx dx by the since ktl 23.
0< diel, NTS o<detiel
Since ktl z3.
Sequente
lby fhe assumpn Pli)
Sine K, K-IZ K, 0de, denel
Sime K,
1s true for Lisk.
deti= deide-rハ1
%3D
So
Oc deH El /
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