Explanation Given information: The formula is, 1 1 ⋅ 2 + 1 2 ⋅ 3 + 1 3 ⋅ 4 + … + 1 n ( n + 1 ) = n n + 1 Here, n is the set of all natural numbers. Concept used: For each natural number n , Let P ( n ) be a statement depending on n . Suppose that the following two conditions are satisfied. 1. P ( 1 ) is true. 2. For every natural number k , if P ( k ) is true then P ( k + 1 ) is true. Then P ( n ) is true for all natural numbers n . To apply this principle, there are two steps, 1. Prove that P ( 1 ) is true. 2. Assume that P ( k ) is true and use this assumption to prove that P ( k + 1 ) is true. The assumption that P ( k ) is true is called the induction hypothesis. Proof: The given statement is, P ( n ) = n n + 1 Substitute 1 for n in above equation. P ( 1 ) = 1 1 + 1 = 1 2 Therefore, P ( 1 ) is true. Now, assume that P ( k ) is true, thus induction hypothesis is, 1 1 ⋅ 2 + 1 2 ⋅ 3 + 1 3 ⋅ 4 + … + 1 k ( k + 1 ) = k k + 1 Start from the left hand side of the given formula for P ( k + 1 )

Bundle: College Algebra, 7th + WebAssign Printed Access Card for Stewart/Redlin/Watson's College Algebra, 7th Edition, Single-Term

7th Edition

ISBN: 9781305718944

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 8.5, Problem 12E

To determine

To prove:

That the formula 11⋅2+12⋅3+13⋅4+…+1n(n+1)=nn+1 is true for all natural numbers n.

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Chapter 8.5, Problem 11E

Chapter 8.5, Problem 13E

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Chapter 8 Solutions

Bundle: College Algebra, 7th + WebAssign Printed Access Card for Stewart/Redlin/Watson's College Algebra, 7th Edition, Single-Term

Ch. 8.1 - A sequence is a function whose domain is...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Terms of a Sequence, Find the first four terms and...Ch. 8.1 - Prob. 6E Ch. 8.1 - Terms of a Sequence, Find the first four terms and...Ch. 8.1 - Terms of a Sequence, Find the first four terms and...Ch. 8.1 - Terms of a Sequence, Find the first four terms and...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Recursive Sequences A sequence is defined...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - nth term of a Sequence Find the nth term of a...Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - nth Partial Sum Find the first four partial sums...Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - Prob. 55E Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.1 - Prob. 61E Ch. 8.1 - Prob. 62E Ch. 8.1 - Prob. 63E Ch. 8.1 - Prob. 64E Ch. 8.1 - Prob. 65E Ch. 8.1 - Sigma Notation Write the sum without using sigma...Ch. 8.1 - Prob. 67E Ch. 8.1 - Prob. 68E Ch. 8.1 - Prob. 69E Ch. 8.1 - Prob. 70E Ch. 8.1 - Prob. 71E Ch. 8.1 - Prob. 72E Ch. 8.1 - Prob. 73E Ch. 8.1 - Prob. 74E Ch. 8.1 - Prob. 75E Ch. 8.1 - Prob. 76E Ch. 8.1 - Prob. 77E Ch. 8.1 - Prob. 78E Ch. 8.1 - Population of a City A city was incorporate in...Ch. 8.1 - Prob. 80E Ch. 8.1 - Prob. 81E Ch. 8.1 - Prob. 82E Ch. 8.1 - Prob. 83E Ch. 8.1 - Prob. 84E Ch. 8.1 - Prob. 85E Ch. 8.1 - Prob. 86E Ch. 8.1 - Prob. 87E Ch. 8.1 - Prob. 88E Ch. 8.2 - An arithmetic sequence is a sequence in which the...Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Arithmetic Sequence? Find the first five terms of...Ch. 8.2 - Prob. 29E Ch. 8.2 - Arithmetic Sequence? Find the first five terms of...Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Terms of an Arithmetic Sequence Determine the...Ch. 8.2 - Prob. 45E Ch. 8.2 - Prob. 46E Ch. 8.2 - Prob. 47E Ch. 8.2 - Prob. 48E Ch. 8.2 - Prob. 49E Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 52E Ch. 8.2 - Prob. 53E Ch. 8.2 - Prob. 54E Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.2 - Prob. 57E Ch. 8.2 - Prob. 58E Ch. 8.2 - Prob. 59E Ch. 8.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 8.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 8.2 - Prob. 62E Ch. 8.2 - Prob. 63E Ch. 8.2 - Prob. 64E Ch. 8.2 - Prob. 65E Ch. 8.2 - Prob. 66E Ch. 8.2 - Prob. 67E Ch. 8.2 - Prob. 68E Ch. 8.2 - Prob. 69E Ch. 8.2 - Prob. 70E Ch. 8.2 - Prob. 71E Ch. 8.2 - Prob. 72E Ch. 8.2 - Prob. 73E Ch. 8.2 - Prob. 74E Ch. 8.2 - Prob. 75E Ch. 8.2 - Prob. 76E Ch. 8.2 - Prob. 77E Ch. 8.2 - Prob. 78E Ch. 8.3 - A geometric sequence is a sequence in which the...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - (a) The nth partial sum of a geometric sequence...Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Prob. 40E Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.3 - Prob. 43E Ch. 8.3 - Prob. 44E Ch. 8.3 - Prob. 45E Ch. 8.3 - Prob. 46E Ch. 8.3 - Prob. 47E Ch. 8.3 - Prob. 48E Ch. 8.3 - Prob. 49E Ch. 8.3 - Prob. 50E Ch. 8.3 - Prob. 51E Ch. 8.3 - Prob. 52E Ch. 8.3 - Prob. 53E Ch. 8.3 - Prob. 54E Ch. 8.3 - Prob. 55E Ch. 8.3 - Prob. 56E Ch. 8.3 - Prob. 57E Ch. 8.3 - Prob. 58E Ch. 8.3 - Prob. 59E Ch. 8.3 - Prob. 60E Ch. 8.3 - Prob. 61E Ch. 8.3 - Prob. 62E Ch. 8.3 - Prob. 63E Ch. 8.3 - Prob. 64E Ch. 8.3 - Prob. 65E Ch. 8.3 - Prob. 66E Ch. 8.3 - Prob. 67E Ch. 8.3 - Prob. 68E Ch. 8.3 - Prob. 69E Ch. 8.3 - Prob. 70E Ch. 8.3 - Prob. 71E Ch. 8.3 - Prob. 72E Ch. 8.3 - Prob. 73E Ch. 8.3 - Prob. 74E Ch. 8.3 - Prob. 75E Ch. 8.3 - Prob. 76E Ch. 8.3 - Prob. 77E Ch. 8.3 - Prob. 78E Ch. 8.3 - Prob. 79E Ch. 8.3 - Prob. 80E Ch. 8.3 - Prob. 81E Ch. 8.3 - Prob. 82E Ch. 8.3 - Repeated Decimal Express the repeating decimal as...Ch. 8.3 - Prob. 84E Ch. 8.3 - Prob. 85E Ch. 8.3 - Prob. 86E Ch. 8.3 - Prob. 87E Ch. 8.3 - Prob. 88E Ch. 8.3 - Bouncing Ball A ball is dropped from a height of...Ch. 8.3 - Prob. 90E Ch. 8.3 - Mixing Coolant A truck radiator holds 5 gal and is...Ch. 8.3 - Musical Frequencies The frequencies of musical...Ch. 8.3 - Bouncing Ball A ball is dropped from a height of 9...Ch. 8.3 - Prob. 94E Ch. 8.3 - Prob. 95E Ch. 8.3 - Prob. 96E Ch. 8.3 - Prob. 97E Ch. 8.3 - Prob. 98E Ch. 8.3 - Prob. 99E Ch. 8.3 - Geometry A circular disk of radius R is cut out of...Ch. 8.3 - Geometry A yellow square of side I is divided into...Ch. 8.3 - Prob. 102E Ch. 8.3 - Prob. 103E Ch. 8.3 - Prob. 104E Ch. 8.4 - An annuity is a sum of money that is paid in...Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Annuity Find the amount of an annuity that...Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Saving How much money should be invested every...Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Mortgage What is the monthly payment on a 30-year...Ch. 8.4 - Mortgage what is the monthly payment on a 30-year...Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Mortgage A couple can afford to make a monthly...Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8.4 - Interest Rate An item at a department store is...Ch. 8.4 - Interest Rate A man purchases a $2000 diamond ring...Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - DISCOVR: Amortizing a Mortgage When they bought...Ch. 8.5 - Mathematical induction is a method of proving that...Ch. 8.5 - Prob. 2E Ch. 8.5 - Prob. 3E Ch. 8.5 - Prob. 4E Ch. 8.5 - Prob. 5E Ch. 8.5 - Prob. 6E Ch. 8.5 - Prob. 7E Ch. 8.5 - Prob. 8E Ch. 8.5 - Prob. 9E Ch. 8.5 - Prob. 10E Ch. 8.5 - Prob. 11E Ch. 8.5 - Prob. 12E Ch. 8.5 - Prob. 13E Ch. 8.5 - Prob. 14E Ch. 8.5 - Prob. 15E Ch. 8.5 - Prob. 16E Ch. 8.5 - Prob. 17E Ch. 8.5 - Prob. 18E Ch. 8.5 - Prob. 19E Ch. 8.5 - Prob. 20E Ch. 8.5 - Prob. 21E Ch. 8.5 - Prob. 22E Ch. 8.5 - Proving a Statement Use mathematical induction to...Ch. 8.5 - Prob. 24E Ch. 8.5 - Prob. 25E Ch. 8.5 - Prob. 26E Ch. 8.5 - Prob. 27E Ch. 8.5 - Prob. 28E Ch. 8.5 - Prob. 29E Ch. 8.5 - Prob. 30E Ch. 8.5 - Prob. 31E Ch. 8.5 - Prob. 32E Ch. 8.5 - Prob. 33E Ch. 8.5 - Prob. 34E Ch. 8.5 - Prob. 35E Ch. 8.5 - Prob. 36E Ch. 8.5 - Prob. 37E Ch. 8.6 - An algebraic expression of the form a+b, which...Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - Prob. 15E Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - Prob. 28E Ch. 8.6 - Prob. 29E Ch. 8.6 - Prob. 30E Ch. 8.6 - Term of a Binomial Expansion Find the indicated...Ch. 8.6 - Prob. 32E Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Term of a Binomial Expansion Find the indicated...Ch. 8.6 - Prob. 38E Ch. 8.6 - Prob. 39E Ch. 8.6 - Prob. 40E Ch. 8.6 - Prob. 41E Ch. 8.6 - Prob. 42E Ch. 8.6 - Prob. 43E Ch. 8.6 - Prob. 44E Ch. 8.6 - Prob. 45E Ch. 8.6 - Prob. 46E Ch. 8.6 - Prob. 47E Ch. 8.6 - Prob. 48E Ch. 8.6 - Prob. 49E Ch. 8.6 - Prob. 50E Ch. 8.6 - Prob. 51E Ch. 8.6 - Prob. 52E Ch. 8.6 - Prob. 53E Ch. 8.6 - Prob. 54E Ch. 8.6 - Prob. 55E Ch. 8.6 - Prob. 56E Ch. 8.6 - Prob. 57E Ch. 8.6 - Prob. 58E Ch. 8.6 - Prob. 59E Ch. 8 - Prob. 1CC Ch. 8 - Prob. 2CC Ch. 8 - Prob. 3CC Ch. 8 - Prob. 4CC Ch. 8 - Prob. 5CC Ch. 8 - Prob. 6CC Ch. 8 - Prob. 7CC Ch. 8 - Prob. 8CC Ch. 8 - Prob. 9CC Ch. 8 - Prob. 10CC Ch. 8 - Prob. 11CC Ch. 8 - Prob. 12CC Ch. 8 - Prob. 1E Ch. 8 - Prob. 2E Ch. 8 - Prob. 3E Ch. 8 - Prob. 4E Ch. 8 - Prob. 5E Ch. 8 - Prob. 6E Ch. 8 - Prob. 7E Ch. 8 - Prob. 8E Ch. 8 - Prob. 9E Ch. 8 - Prob. 10E Ch. 8 - Prob. 11E Ch. 8 - Prob. 12E Ch. 8 - Prob. 13E Ch. 8 - Prob. 14E Ch. 8 - Arithmetic or Geometric? The first four terms of a...Ch. 8 - Prob. 16E Ch. 8 - Prob. 17E Ch. 8 - Prob. 18E Ch. 8 - Prob. 19E Ch. 8 - Prob. 20E Ch. 8 - Prob. 21E Ch. 8 - Prob. 22E Ch. 8 - Prob. 23E Ch. 8 - Prob. 24E Ch. 8 - Prob. 25E Ch. 8 - Prob. 26E Ch. 8 - Prob. 27E Ch. 8 - Finding Terms of Arithmetic and Geometric...Ch. 8 - Salary A teacher makes $32,000 in his first year...Ch. 8 - SalaryA Colleague of the teacher in Exercise 29,...Ch. 8 - Prob. 31E Ch. 8 - Prob. 32E Ch. 8 - Prob. 33E Ch. 8 - Prob. 34E Ch. 8 - Prob. 35E Ch. 8 - Prob. 36E Ch. 8 - Prob. 37E Ch. 8 - Prob. 38E Ch. 8 - Prob. 39E Ch. 8 - Prob. 40E Ch. 8 - Prob. 41E Ch. 8 - Prob. 42E Ch. 8 - Prob. 43E Ch. 8 - Prob. 44E Ch. 8 - Prob. 45E Ch. 8 - Prob. 46E Ch. 8 - Prob. 47E Ch. 8 - Prob. 48E Ch. 8 - Prob. 49E Ch. 8 - Prob. 50E Ch. 8 - Prob. 51E Ch. 8 - Prob. 52E Ch. 8 - Prob. 53E Ch. 8 - Prob. 54E Ch. 8 - Prob. 55E Ch. 8 - Prob. 56E Ch. 8 - Prob. 57E Ch. 8 - Prob. 58E Ch. 8 - Prob. 59E Ch. 8 - Prob. 60E Ch. 8 - Prob. 61E Ch. 8 - Prob. 62E Ch. 8 - AncestorsA person has two parents, four...Ch. 8 - Prob. 64E Ch. 8 - Prob. 65E Ch. 8 - Prob. 66E Ch. 8 - Prob. 67E Ch. 8 - Prob. 68E Ch. 8 - Prob. 69E Ch. 8 - Prob. 70E Ch. 8 - Prob. 71E Ch. 8 - Prob. 72E Ch. 8 - Prob. 73E Ch. 8 - Prob. 74E Ch. 8 - Prob. 75E Ch. 8 - Prob. 76E Ch. 8 - Prob. 77E Ch. 8 - Prob. 78E Ch. 8 - Prob. 79E Ch. 8 - Prob. 80E Ch. 8 - Prob. 81E Ch. 8 - Prob. 82E Ch. 8 - Prob. 83E Ch. 8 - Prob. 1T Ch. 8 - Prob. 2T Ch. 8 - Prob. 3T Ch. 8 - Prob. 4T Ch. 8 - Prob. 5T Ch. 8 - Prob. 6T Ch. 8 - Prob. 7T Ch. 8 - Prob. 8T Ch. 8 - Prob. 9T Ch. 8 - Prob. 10T Ch. 8 - Prob. 11T Ch. 8 - Prob. 12T Ch. 8 - Prob. 13T Ch. 8 - Prob. 1P Ch. 8 - Fitness Progam Sheila decides to embark on a...Ch. 8 - Monthly Savings Program Alice opens a savings...Ch. 8 - Pollution A chemical plant discharges 2400 tons of...Ch. 8 - Prob. 5P Ch. 8 - Prob. 6P

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To prove: That the formula 1 1 ⋅ 2 + 1 2 ⋅ 3 + 1 3 ⋅ 4 + … + 1 n ( n + 1 ) = n n + 1 is true for all natural numbers n .

Solution Summary: The author explains the induction hypothesis, which assumes that P(1) is true for all natural numbers.

Videos

To prove:

That the formula 11⋅2+12⋅3+13⋅4+…+1n(n+1)=nn+1 is true for all natural numbers n.

Want to see the full answer?

Chapter 8 Solutions

To prove: That the formula 1 1 ⋅ 2 + 1 2 ⋅ 3 + 1 3 ⋅ 4 + … + 1 n ( n + 1 ) = n n + 1 is true for all natural numbers n .

Solution Summary: The author explains the induction hypothesis, which assumes that P(1) is true for all natural numbers.

Videos

To prove: That the formula 11⋅2+12⋅3+13⋅4+…+1n(n+1)=nn+1 is true for all natural numbers n.

Want to see the full answer?

Chapter 8 Solutions

To prove:

That the formula 11⋅2+12⋅3+13⋅4+…+1n(n+1)=nn+1 is true for all natural numbers n.