162 Chapter 6.13. Prove that1-142-2! +..+n-n! = (n+ 1)! - 1 for every positive integern. 6.14. Prove that 2! - 4! - 6! ..... (2n)! > [(n + 1)!]" for every positive integer n. 6.15. Prove that ++ ++..++ <2n -1 for every positive integer n. 6.16. Prove that 7| [34+1 -52-1] for every positive integer n.
162 Chapter 6.13. Prove that1-142-2! +..+n-n! = (n+ 1)! - 1 for every positive integern. 6.14. Prove that 2! - 4! - 6! ..... (2n)! > [(n + 1)!]" for every positive integer n. 6.15. Prove that ++ ++..++ <2n -1 for every positive integer n. 6.16. Prove that 7| [34+1 -52-1] for every positive integer n.
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Author:Erwin Kreyszig
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Help me with 6.14
![162
Chapter 6 Mathematical Induction
6.13. Prove that 1 · 1!+ 2 · 2! + •.. +n.n! = (n+ 1)! - 1 for every positive integer n,
6.14. Prove that 2! · 4! - 6! . .... (2n)! > [(n + 1)!]" for every positive integer n.
6.15. Prove that + ++++ t <2ñ - 1 for every positive integer n.
6.16. Prove that 7| [34n+1 _ 52n-1] for every positive integer n.
The
Principle a
Mathematic
0s Jon zoob doid
ןuקחכנ ןכ
.olenimonob ed
To buoAG
6.2 A MORE GENERAL PRINCIPLE
OF MATHEMATICAL INDUCTION
The Principle of Mathematical Induction, described in the preceding section, gives ..
technique for proving that a statement of the type
bobo.ilow.ous nivollo odi lo
For every positive integer n, P(n).
Resu
is true. There are situations, however, when the domain of P(n) consists of those intega
greater than or equal to some fixed integer m different from 1. We now describe u
analogous technique to verify the truth of a statement of the following type where i
denotes some fixed integer:
ucibie o
n
oln ei odi Ao onion s 21b 21dun Iss lo oe borobio llow yne zi Ae
For every integer n > m, P(n).
ordered, that is, every nonempty subset of N has a least element. As a
the Well-Ordering Principle, other sets are also well-ordered.
101
Theorem 6.7 For each integer m, the set
PROOF
2++gaibbe
) S= {i e Z: i>m}
is well-ordered.
ww.pto 1odm1un h
ieste comborcq
vitizog u
no svad oW (
i>m}. For each integer n e S, let P(n) be :
d alutnol 0
(1) P(m) i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F21ee6c6c-c0d0-45fb-84eb-2197ba19fad3%2Fg6gq2er_processed.jpeg&w=3840&q=75)
Transcribed Image Text:162
Chapter 6 Mathematical Induction
6.13. Prove that 1 · 1!+ 2 · 2! + •.. +n.n! = (n+ 1)! - 1 for every positive integer n,
6.14. Prove that 2! · 4! - 6! . .... (2n)! > [(n + 1)!]" for every positive integer n.
6.15. Prove that + ++++ t <2ñ - 1 for every positive integer n.
6.16. Prove that 7| [34n+1 _ 52n-1] for every positive integer n.
The
Principle a
Mathematic
0s Jon zoob doid
ןuקחכנ ןכ
.olenimonob ed
To buoAG
6.2 A MORE GENERAL PRINCIPLE
OF MATHEMATICAL INDUCTION
The Principle of Mathematical Induction, described in the preceding section, gives ..
technique for proving that a statement of the type
bobo.ilow.ous nivollo odi lo
For every positive integer n, P(n).
Resu
is true. There are situations, however, when the domain of P(n) consists of those intega
greater than or equal to some fixed integer m different from 1. We now describe u
analogous technique to verify the truth of a statement of the following type where i
denotes some fixed integer:
ucibie o
n
oln ei odi Ao onion s 21b 21dun Iss lo oe borobio llow yne zi Ae
For every integer n > m, P(n).
ordered, that is, every nonempty subset of N has a least element. As a
the Well-Ordering Principle, other sets are also well-ordered.
101
Theorem 6.7 For each integer m, the set
PROOF
2++gaibbe
) S= {i e Z: i>m}
is well-ordered.
ww.pto 1odm1un h
ieste comborcq
vitizog u
no svad oW (
i>m}. For each integer n e S, let P(n) be :
d alutnol 0
(1) P(m) i
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