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Proof by Induction
Synopsis
This series of booklets is designed to give a thorough grounding in Inductive Proof. A good understanding in this topic is essential for all those people who wish to go on and study Mathematics and other science subjects like, Physics, Chemistry and Engineering.
Often the fundamental concepts like these are frequently not completely understood leading to confusion and often dissatisfaction and can result in people giving up on the subject altogether.
This is a shame because such confusion can often be attributed to poor teaching or unexplained misconceptions. This series of booklets sets out to explain fully and clearly every step throughout with everything presented in an easy to understand format.
There are a sufficient number of examples in this booklet to cover most types of questions that the student is likely to encounter.
All booklets in this series are competitively priced
This booklet is Proof by Induction IP1 in the series and contains the following
Prove that 1+2+3+ . . . + n = ½ n (n+1) where n ≥ 1
Prove that for any integer n ≥ 1 +. . .+ (2n-1) = n²
Prove that for any integer n ≥ 1 . . . +2n = n (1 + l) – n2
Prove that n !≤ n² for any integer n≥1
Prove that 2n > n2 for any integer ≥ 5
Prove that for n ≥ 1; 2n3 -3n2 + n + 31 ≥ 0
Prove that 4n > n4 for any integer n >4
Prove that 12+22+32 +. . . +n2 = n(n +1)(2n +1)/6 where n ≥1
Prove that 13+23+33+ . . . +n3 = ¼ n2 (n+1)2 for n≥1
Prove that 1⁄1.2+1⁄2.3+1⁄3.4+ . . .+ 1⁄n( n+1) = n⁄( n+1)
Prove that 22+42+62+. . .+(2n)2 = 2⁄3n(n+1)(2n+1)
Prove that 1.4+2.5+3.6+ . . . +n(n+3) = n(n+1)(n+5) ⁄3
Prove that 23+43+63+ . . .+ (2n)3 = 2n2(n+1)2 where n ≥1.
Prove that 4 + 42 + 43 + . . . + 4n = 4⁄3 (4n – 1) Where n ≥ 1
Prove that (2+3n) = n⁄2 (3n +7) where n ≥1
Prove that 3 divides into 4n+5 where n ≥ 1
Prove that n!>2n where n ≥ 4
Prove that 23+43+63+. . . +(2n)3 = 2n2(n+1)2 Where n ≥ 1
Prove that – 1 +4 –8 + . . . +(–1)nn2 = 1⁄2 (–1)nn(n+1) Where n ≥ 1
Prove (1– 1⁄22 )(1– 1⁄32) (1–1⁄42) 𝖷 ... 𝖷 (1– 1⁄n2) = n+1⁄2n where n ∈ ℤ+ and n ≥ 2
Prove that for x > – 1 and n ≥ 1 that (1 + x)n ≥ 1 + nx
Prove that 7|8n–1 for n ≥ 1
Prove that 5|42n – 1
Prove that 4|3(72n – 1) for n ≥ 1
Prove that for n ≥1 (1)! + 2(2)! + 3(3)! + . . . + n(n)! = (n+1)! – 1
Prove that (ab)n = a nbn for every natural number n
25 fully worked solutions
No of 22
Other booklets in this series will be published shortly so keep a lookout for these as they become available.
This series of booklets is designed to give a thorough grounding in Inductive Proof. A good understanding in this topic is essential for all those people who wish to go on and study Mathematics and other science subjects like, Physics, Chemistry and Engineering.
Often the fundamental concepts like these are frequently not completely understood leading to confusion and often dissatisfaction and can result in people giving up on the subject altogether.
This is a shame because such confusion can often be attributed to poor teaching or unexplained misconceptions. This series of booklets sets out to explain fully and clearly every step throughout with everything presented in an easy to understand format.
There are a sufficient number of examples in this booklet to cover most types of questions that the student is likely to encounter.
All booklets in this series are competitively priced
This booklet is Proof by Induction IP1 in the series and contains the following
Prove that 1+2+3+ . . . + n = ½ n (n+1) where n ≥ 1
Prove that for any integer n ≥ 1 +. . .+ (2n-1) = n²
Prove that for any integer n ≥ 1 . . . +2n = n (1 + l) – n2
Prove that n !≤ n² for any integer n≥1
Prove that 2n > n2 for any integer ≥ 5
Prove that for n ≥ 1; 2n3 -3n2 + n + 31 ≥ 0
Prove that 4n > n4 for any integer n >4
Prove that 12+22+32 +. . . +n2 = n(n +1)(2n +1)/6 where n ≥1
Prove that 13+23+33+ . . . +n3 = ¼ n2 (n+1)2 for n≥1
Prove that 1⁄1.2+1⁄2.3+1⁄3.4+ . . .+ 1⁄n( n+1) = n⁄( n+1)
Prove that 22+42+62+. . .+(2n)2 = 2⁄3n(n+1)(2n+1)
Prove that 1.4+2.5+3.6+ . . . +n(n+3) = n(n+1)(n+5) ⁄3
Prove that 23+43+63+ . . .+ (2n)3 = 2n2(n+1)2 where n ≥1.
Prove that 4 + 42 + 43 + . . . + 4n = 4⁄3 (4n – 1) Where n ≥ 1
Prove that (2+3n) = n⁄2 (3n +7) where n ≥1
Prove that 3 divides into 4n+5 where n ≥ 1
Prove that n!>2n where n ≥ 4
Prove that 23+43+63+. . . +(2n)3 = 2n2(n+1)2 Where n ≥ 1
Prove that – 1 +4 –8 + . . . +(–1)nn2 = 1⁄2 (–1)nn(n+1) Where n ≥ 1
Prove (1– 1⁄22 )(1– 1⁄32) (1–1⁄42) 𝖷 ... 𝖷 (1– 1⁄n2) = n+1⁄2n where n ∈ ℤ+ and n ≥ 2
Prove that for x > – 1 and n ≥ 1 that (1 + x)n ≥ 1 + nx
Prove that 7|8n–1 for n ≥ 1
Prove that 5|42n – 1
Prove that 4|3(72n – 1) for n ≥ 1
Prove that for n ≥1 (1)! + 2(2)! + 3(3)! + . . . + n(n)! = (n+1)! – 1
Prove that (ab)n = a nbn for every natural number n
25 fully worked solutions
No of 22
Other booklets in this series will be published shortly so keep a lookout for these as they become available.
23 pages, Kindle Edition
First published September 23, 2014
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Displaying 1 of 1 review
December 16, 2018
Ok
It was helpful.
It was helpful. I have not gotten a chance to read the whole book. When I do I shall give a better review.
It was helpful.
It was helpful. I have not gotten a chance to read the whole book. When I do I shall give a better review.
Displaying 1 of 1 review