Prove that Va ЄR, Vn ЄN, [0 < a < 1] ⇒ a" ≤1 using mathematical induction. Justify every step.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter 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)
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.)
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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