Prove that Va ЄR, Vn ЄN, [0 < a < 1] ⇒ a" ≤1 using mathematical induction. Justify every step.)
Prove that Va ЄR, Vn ЄN, [0 < a < 1] ⇒ a" ≤1 using mathematical induction. Justify every step.)
Chapter8: Sequences, Series,and Probability
Section: Chapter Questions
Problem 32CT
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Please help me with these questions. I am having trouble understanding what to do. Please check if both of the questions with answer are correct.(The correct steps + answers are shown/ proven). If not please show the correct answers to the questions
Image 1: The two questions
Image 2: Answers to each of the two questions
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![Ex # 1x3
Va e R, Vne N, CO < a < 1] - an≤1
→ For n = 1
104941-Da≤ 1, which is true
n
For n= k, the statement is true c
Va e R, VK & IN, LOLa<1] -Dak≤ 1
-> We prove that it is true for n = k +1 also org
ط
a
k+1 = aka
IN
.
ak +1
41
We have,
(x(+2)
ak
and as 1. So their product
is also
(1+2)
202
So, by principle of mathematical
induction, Ya ER, VD & IN,
[o<<]] D an IV. (Proved).ton
Ex # 2 3
#2
IN, [n >2 ->n! <n²]-[120
[n>2^n! <<n²]-[Y
IN,
Vne
-> For n=3,
n! = 3! = 6
n
D² = 33 = 27
6<27. it is true tomato sot
→ Let the statement is true for n=k i.e, k! <KK
k sum
-> Now, we will prove for n = +1 band song We
LHS = (K+1)!
= (K + 1)K!
< (k + 1) KK
SW
< (k+1) (K+1) Koo
(Since, K < K+1
RHS = (K+1) K+1
So, KK < (1+1) K
So, (k+1)! <(K+1) K+1
Hence, by mathematical induction, Vn e IN, En22-on!<n²]
(Proved)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9d632c5-d4ea-478f-a512-43d3c8c76b91%2F68c97180-b2bd-4c52-b670-3042dc1c3aff%2Fzbfg4x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ex # 1x3
Va e R, Vne N, CO < a < 1] - an≤1
→ For n = 1
104941-Da≤ 1, which is true
n
For n= k, the statement is true c
Va e R, VK & IN, LOLa<1] -Dak≤ 1
-> We prove that it is true for n = k +1 also org
ط
a
k+1 = aka
IN
.
ak +1
41
We have,
(x(+2)
ak
and as 1. So their product
is also
(1+2)
202
So, by principle of mathematical
induction, Ya ER, VD & IN,
[o<<]] D an IV. (Proved).ton
Ex # 2 3
#2
IN, [n >2 ->n! <n²]-[120
[n>2^n! <<n²]-[Y
IN,
Vne
-> For n=3,
n! = 3! = 6
n
D² = 33 = 27
6<27. it is true tomato sot
→ Let the statement is true for n=k i.e, k! <KK
k sum
-> Now, we will prove for n = +1 band song We
LHS = (K+1)!
= (K + 1)K!
< (k + 1) KK
SW
< (k+1) (K+1) Koo
(Since, K < K+1
RHS = (K+1) K+1
So, KK < (1+1) K
So, (k+1)! <(K+1) K+1
Hence, by mathematical induction, Vn e IN, En22-on!<n²]
(Proved)
![1. Prove that
VaR, VnEN, [0 < a < 1] ⇒ a" ≤1
using mathematical induction. Justify every step.)
2. Prove that
Vn N, [n>2n!<n"]
using mathematical induction. Justify every step.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9d632c5-d4ea-478f-a512-43d3c8c76b91%2F68c97180-b2bd-4c52-b670-3042dc1c3aff%2Fz2n972j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove that
VaR, VnEN, [0 < a < 1] ⇒ a" ≤1
using mathematical induction. Justify every step.)
2. Prove that
Vn N, [n>2n!<n"]
using mathematical induction. Justify every step.)
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