1. Prove that Va € R, Vn € N, [0 < a <1] ⇒ a" ≤1 using mathematical induction. Justify every step.) 2. Prove that VnEN, [n>2n!
1. Prove that Va € R, Vn € N, [0 < a <1] ⇒ a" ≤1 using mathematical induction. Justify every step.) 2. Prove that VnEN, [n>2n!
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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Image 1: The 2 Questions 1 &2
Image 2: The Answers to the question (top part of the image is the answer to question #1, while the botton is answer to question #2)
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![1. Prove that
Va ER, Vn Є N, [0 < a < 1] ⇒ an ≤1
using mathematical induction. Justify every step.)
2. Prove that
Vn EN, [n>2n!<n"]
using mathematical induction. Justify every step.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bb68d7e-0dda-4f44-bcde-638a355d6dc4%2F9a561ca0-3fce-4d2c-bcab-6444386c18fd%2Fxk9byfq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove that
Va ER, Vn Є N, [0 < a < 1] ⇒ an ≤1
using mathematical induction. Justify every step.)
2. Prove that
Vn EN, [n>2n!<n"]
using mathematical induction. Justify every step.)
![Step 1: Solving 1
Base Step
For
n21.
029<1
Let
for
=> a ≤1, which is true.
n=k,
the statement is true.
HaER, NKEN, [oca<1] = a* ≤1.
We
prove
a
that it is true for n=k+1 also.
k+ 1 K. a.
=
≤ 1
a
K+1 ≤ 1
к
a
We have,
K
and a≤1.
product
So, their
is
also ≤1
So, by principle of mathematical
induction, HaER, In EN,
[o ca<] a" ≤1.
Step 2: Solving 2
2
For
n = 3
n! = 3! = 6
nn = 3³ = 27.
6 < 27.
it is true.
Let
the
statement is true
1
Now, we
will
prove for nak+1.
for nak ie,
K
k! <K,
LHS = (k+1)!
=
(K+D)K!
< (k+1) K
< (K+D(k+1)*
501
1
(since, k<k+1
<(x+1)").
(K+D!
(K+DK+1 = RHS.
(k+1)+1.
So, K
Hence, by induction, the N, [n> 2 >n! <n^].
(Proved).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bb68d7e-0dda-4f44-bcde-638a355d6dc4%2F9a561ca0-3fce-4d2c-bcab-6444386c18fd%2Fkdxo3gc_processed.png&w=3840&q=75)
Transcribed Image Text:Step 1: Solving 1
Base Step
For
n21.
029<1
Let
for
=> a ≤1, which is true.
n=k,
the statement is true.
HaER, NKEN, [oca<1] = a* ≤1.
We
prove
a
that it is true for n=k+1 also.
k+ 1 K. a.
=
≤ 1
a
K+1 ≤ 1
к
a
We have,
K
and a≤1.
product
So, their
is
also ≤1
So, by principle of mathematical
induction, HaER, In EN,
[o ca<] a" ≤1.
Step 2: Solving 2
2
For
n = 3
n! = 3! = 6
nn = 3³ = 27.
6 < 27.
it is true.
Let
the
statement is true
1
Now, we
will
prove for nak+1.
for nak ie,
K
k! <K,
LHS = (k+1)!
=
(K+D)K!
< (k+1) K
< (K+D(k+1)*
501
1
(since, k<k+1
<(x+1)").
(K+D!
(K+DK+1 = RHS.
(k+1)+1.
So, K
Hence, by induction, the N, [n> 2 >n! <n^].
(Proved).
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