Explanation Theorem used: Binomial Theorem “Let x and a be real numbers. For any positive integer n , ( x + a ) n = ( n 0 ) x n + ( n 1 ) a x n − 1 + ( n 2 ) a 2 x n − 2 + ⋯ ( n j ) a j x n − j + ⋯ ( n n ) a n Here, the symbol ( n j ) is said to be a binomial coefficient”. Formula used: “If j and n are integers with 0 ≤ j ≤ n , then ( n j ) = n ! j ! ( n − j ) ! ”. Property used: The property of factorial is n ! = n ( n − 1 ) ! if n ≥ 2 . Calculation: It is given that, H ( x ) = x 5 − 4 x 4 and H ( p + 1 ) = ( p + 1 ) 5 − 4 ( p + 1 ) 4 . Find the expansion of ( p + 1 ) 5 and ( p + 1 ) 4 . Compare the expression ( p + 1 ) 5 with ( x + a ) n , the value of n = 5 and x = z a = 1. Use the Binomial Theorem to expand the binomial ( p + 1 ) 5 as follows. ( p + 1 ) 5 = ( 5 0 ) ( p ) 5 + ( 5 1 ) ( 1 ) ( p ) 4 + ( 5 2 ) ( 1 ) 2 ( p ) 3 + ( 5 3 ) ( 1 ) 3 ( p ) 2 + ( 5 4 ) ( 1 ) 4 ( p ) 1 + ( 5 5 ) ( 1 ) 5 = ( 5 0 ) ( p 5 ) + ( 5 1 ) ( p 4 ) + ( 5 2 ) ( p 3 ) + ( 5 3 ) ( p 2 ) + ( 5 4 ) ( p ) + ( 5 5 ) Apply the above stated formula to find the value of the binomial coefficients as follows. ( p + 1 ) 5 = [ 5 ! 0 ! ( 5 − 0 ) ! p 5 + 5 ! 1 ! ( 5 − 1 ) ! p 4 + 5 ! 2 ! ( 5 − 2 ) ! p 3 + 5 ! 3 ! ( 5 − 3 ) ! p 2 + 5 ! 4 ! ( 5 − 4 ) ! p + 5 ! 5 ! ( 5 − 5 ) ! ] = 5 ! 0 ! ⋅ 5 ! p 5 + 5 ! 1 ! ⋅ 4 ! p 4 + 5 ! 2 ! ⋅ 3 ! p 3 + 5 ! 3 ! ⋅ 2 ! p 2 + 5 ! 4 ! ⋅ 1 ! p + 5 ! 5 ! ⋅ 0 ! = 5 ! 5 ! p 5 + 5 ! 4 ! p 4 + 5 ! 2 ! ⋅ 3 ! p 3 + 5 ! 3 ! ⋅ 2 ! p 2 + 5 ! 4 ! p + 5 ! 5 ! [ 0 ! = 1 ; 1 ! = 1 ] Use the property of factorial and simplify further. ( p + 1 ) 5 = 5 ! 5 ! p 5 + 5 ⋅ 4 ! 4 ! p 4 + 5 ⋅ 4 ⋅ 3 ! 2 ⋅ 1 ⋅ 3 ! p 3 + 5 ⋅ 4 ⋅ 3 ! 3 ! ⋅ 2 ⋅ 1 p 2 + 5 ⋅ 4 ! 4 ! p + 5 ! 5 ! = 1 ⋅ p 5 + 5 ⋅ p 4 + 10 ⋅ p 3 + 10 ⋅ p 2 + 5 ⋅ p + 1 = p 5 + 5 p 4 + 10 p 3 + 10 p 2 + 5 p + 1 Thus, the expansion for ( p + 1 ) 5 = p 5 + 5 p 4 + 10 p 3 + 10 p 2 + 5 p + 1

4th Edition

ISBN: 9780134556079

Author: Sullivan, Michael, III, Struve, Katherine R., Mazzarella, Janet.

Publisher: Pearson,

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Chapter 13.4, Problem 55E

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To find: The expansion of H(p+1) using the Binomial Theorem.

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Chapter 13.4, Problem 54E

Chapter 13.4, Problem 56E

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Chapter 13 Solutions

Elementary & Intermediate Algebra

Ch. 13.1 - A(n) _____ is a function whose domain is the set...Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - Prob. 15E Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Prob. 18E Ch. 13.1 - Prob. 19E Ch. 13.1 - Prob. 20E Ch. 13.1 - Prob. 21E Ch. 13.1 - Prob. 22E Ch. 13.1 - Prob. 23E Ch. 13.1 - Prob. 24E Ch. 13.1 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Prob. 35E Ch. 13.1 - Prob. 36E Ch. 13.1 - Prob. 37E Ch. 13.1 - Prob. 38E Ch. 13.1 - Prob. 39E Ch. 13.1 - Prob. 40E Ch. 13.1 - Prob. 41E Ch. 13.1 - Prob. 42E Ch. 13.1 - Prob. 43E Ch. 13.1 - Prob. 44E Ch. 13.1 - Prob. 45E Ch. 13.1 - Prob. 46E Ch. 13.1 - Prob. 47E Ch. 13.1 - Prob. 48E Ch. 13.1 - Prob. 49E Ch. 13.1 - Prob. 50E Ch. 13.1 - Prob. 51E Ch. 13.1 - Prob. 52E Ch. 13.1 - Prob. 53E Ch. 13.1 - Prob. 54E Ch. 13.1 - Prob. 55E Ch. 13.1 - Population Growth According to the United States...Ch. 13.1 - Prob. 57E Ch. 13.1 - Prob. 58E Ch. 13.1 - Prob. 59E Ch. 13.1 - Prob. 60E Ch. 13.1 - Prob. 61E Ch. 13.1 - In Problems 5962, a sequence is defined...Ch. 13.1 - Prob. 63E Ch. 13.1 - Prob. 65E Ch. 13.1 - Prob. 66E Ch. 13.1 - Prob. 67E Ch. 13.1 - Prob. 68E Ch. 13.1 - Prob. 69E Ch. 13.1 - Prob. 70E Ch. 13.1 - Prob. 71E Ch. 13.1 - Prob. 72E Ch. 13.1 - Prob. 73E Ch. 13.1 - Prob. 74E Ch. 13.1 - Prob. 75E Ch. 13.1 - Prob. 76E Ch. 13.1 - Prob. 77E Ch. 13.1 - Prob. 78E Ch. 13.1 - Prob. 79E Ch. 13.1 - Prob. 80E Ch. 13.1 - Prob. 81E Ch. 13.1 - Prob. 82E Ch. 13.1 - Prob. 83E Ch. 13.1 - Prob. 84E Ch. 13.1 - Prob. 85E Ch. 13.1 - Prob. 86E Ch. 13.1 - Prob. 87E Ch. 13.1 - Use technology to find the sum. k=18[(1)kk]Ch. 13.2 - In a(n) _______ sequence, the difference between...Ch. 13.2 - Prob. 2E Ch. 13.2 - Prob. 3E Ch. 13.2 - Prob. 4E Ch. 13.2 - Prob. 5E Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 7E Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Prob. 13E Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Prob. 21E Ch. 13.2 - Prob. 22E Ch. 13.2 - Prob. 23E Ch. 13.2 - Prob. 24E Ch. 13.2 - Prob. 25E Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Prob. 30E Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Prob. 35E Ch. 13.2 - Prob. 36E Ch. 13.2 - Prob. 37E Ch. 13.2 - Prob. 38E Ch. 13.2 - Prob. 39E Ch. 13.2 - Prob. 40E Ch. 13.2 - Prob. 41E Ch. 13.2 - Prob. 42E Ch. 13.2 - Prob. 43E Ch. 13.2 - Prob. 44E Ch. 13.2 - Prob. 45E Ch. 13.2 - Prob. 46E Ch. 13.2 - Prob. 47E Ch. 13.2 - Find the sum of the first 75 terms of the sequence...Ch. 13.2 - Prob. 49E Ch. 13.2 - Prob. 50E Ch. 13.2 - Prob. 51E Ch. 13.2 - Prob. 52E Ch. 13.2 - Prob. 53E Ch. 13.2 - Prob. 54E Ch. 13.2 - Prob. 55E Ch. 13.2 - Prob. 56E Ch. 13.2 - Prob. 57E Ch. 13.2 - Prob. 58E Ch. 13.2 - Prob. 59E Ch. 13.2 - Prob. 60E Ch. 13.2 - Prob. 61E Ch. 13.2 - Prob. 62E Ch. 13.2 - Prob. 63E Ch. 13.2 - Prob. 64E Ch. 13.2 - Prob. 65E Ch. 13.2 - Prob. 66E Ch. 13.2 - Prob. 67E Ch. 13.2 - Prob. 68E Ch. 13.2 - Prob. 69E Ch. 13.2 - Prob. 70E Ch. 13.2 - Prob. 71E Ch. 13.2 - Prob. 72E Ch. 13.2 - Prob. 73E Ch. 13.2 - Compute f(2)f(1), f(3)f(2), and f(4)f(3) for each...Ch. 13.2 - In Problems 75-78, use technology to find the sum...Ch. 13.2 - Prob. 76E Ch. 13.2 - Prob. 77E Ch. 13.2 - In Problems 75-78, use technology to find the sum...Ch. 13.3 - In a(n) __ sequence, the difference between...Ch. 13.3 - Prob. 2E Ch. 13.3 - Prob. 3E Ch. 13.3 - Prob. 4E Ch. 13.3 - Prob. 5E Ch. 13.3 - In Problems 5-7, determine whether the sequence is...Ch. 13.3 - Prob. 7E Ch. 13.3 - Prob. 8E Ch. 13.3 - Prob. 9E Ch. 13.3 - Prob. 10E Ch. 13.3 - Prob. 11E Ch. 13.3 - Prob. 12E Ch. 13.3 - Prob. 13E Ch. 13.3 - Prob. 14E Ch. 13.3 - Prob. 15E Ch. 13.3 - Express 0.2 as a fraction in lowest terms.Ch. 13.3 - Redo Example 9 if the marginal propensity to...Ch. 13.3 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Prob. 21E Ch. 13.3 - Prob. 22E Ch. 13.3 - Prob. 23E Ch. 13.3 - Prob. 24E Ch. 13.3 - Prob. 25E Ch. 13.3 - Prob. 26E Ch. 13.3 - Prob. 27E Ch. 13.3 - Prob. 28E Ch. 13.3 - Prob. 29E Ch. 13.3 - Prob. 30E Ch. 13.3 - Prob. 31E Ch. 13.3 - Prob. 32E Ch. 13.3 - Prob. 33E Ch. 13.3 - Prob. 34E Ch. 13.3 - Prob. 35E Ch. 13.3 - Prob. 36E Ch. 13.3 - Prob. 37E Ch. 13.3 - In Problems 3540, find the indicated term of each...Ch. 13.3 - Prob. 39E Ch. 13.3 - Prob. 40E Ch. 13.3 - Prob. 41E Ch. 13.3 - Prob. 42E Ch. 13.3 - Prob. 43E Ch. 13.3 - Prob. 44E Ch. 13.3 - Prob. 45E Ch. 13.3 - Prob. 46E Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E Ch. 13.3 - Prob. 49E Ch. 13.3 - Prob. 50E Ch. 13.3 - Prob. 51E Ch. 13.3 - Prob. 52E Ch. 13.3 - Prob. 53E Ch. 13.3 - Prob. 54E Ch. 13.3 - Prob. 55E Ch. 13.3 - Prob. 56E Ch. 13.3 - Prob. 57E Ch. 13.3 - Prob. 58E Ch. 13.3 - Prob. 59E Ch. 13.3 - Prob. 60E Ch. 13.3 - Prob. 61E Ch. 13.3 - Prob. 62E Ch. 13.3 - Prob. 63E Ch. 13.3 - Prob. 64E Ch. 13.3 - Prob. 65E Ch. 13.3 - Prob. 66E Ch. 13.3 - Prob. 67E Ch. 13.3 - Prob. 68E Ch. 13.3 - Prob. 69E Ch. 13.3 - Prob. 70E Ch. 13.3 - Prob. 71E Ch. 13.3 - Prob. 72E Ch. 13.3 - Prob. 73E Ch. 13.3 - Prob. 74E Ch. 13.3 - Prob. 75E Ch. 13.3 - Prob. 76E Ch. 13.3 - Prob. 77E Ch. 13.3 - Prob. 78E Ch. 13.3 - Prob. 79E Ch. 13.3 - Prob. 80E Ch. 13.3 - Prob. 81E Ch. 13.3 - Bouncing Balls A ball is dropped from a height of...Ch. 13.3 - Prob. 83E Ch. 13.3 - Prob. 84E Ch. 13.3 - Prob. 85E Ch. 13.3 - Prob. 86E Ch. 13.3 - Prob. 87E Ch. 13.3 - Prob. 88E Ch. 13.3 - Prob. 89E Ch. 13.3 - Prob. 90E Ch. 13.3 - Prob. 91E Ch. 13.3 - Prob. 92E Ch. 13.3 - Prob. 93E Ch. 13.3 - Prob. 94E Ch. 13.3 - Prob. 95E Ch. 13.3 - Prob. 96E Ch. 13.3 - Prob. 97E Ch. 13.3 - Prob. 98E Ch. 13.3 - Prob. 100E Ch. 13.3 - Prob. 101E Ch. 13.3 - Prob. 102E Ch. 13.3 - What is 13+23? Now add the repeating decimals...Ch. 13.3 - Prob. 104E Ch. 13.3 - Prob. 105E Ch. 13.3 - Prob. 106E Ch. 13.3 - Prob. 107E Ch. 13.3 - Prob. 108E Ch. 13.4 - If n 2 is an integer, then n! =...Ch. 13.4 - Prob. 2E Ch. 13.4 - Prob. 3E Ch. 13.4 - Prob. 4E Ch. 13.4 - Prob. 5E Ch. 13.4 - Prob. 6E Ch. 13.4 - Prob. 7E Ch. 13.4 - Prob. 8E Ch. 13.4 - Prob. 9E Ch. 13.4 - Prob. 10E Ch. 13.4 - In Problems 11-18, evalute each factorial...Ch. 13.4 - Prob. 12E Ch. 13.4 - Prob. 13E Ch. 13.4 - Prob. 14E Ch. 13.4 - Prob. 15E Ch. 13.4 - Prob. 16E Ch. 13.4 - Prob. 17E Ch. 13.4 - Prob. 18E Ch. 13.4 - Prob. 19E Ch. 13.4 - Prob. 20E Ch. 13.4 - Prob. 21E Ch. 13.4 - Prob. 22E Ch. 13.4 - Prob. 23E Ch. 13.4 - Prob. 24E Ch. 13.4 - Prob. 25E Ch. 13.4 - Prob. 26E Ch. 13.4 - Prob. 27E Ch. 13.4 - Prob. 28E Ch. 13.4 - Prob. 29E Ch. 13.4 - In Problems 23-38, expand each expression using...Ch. 13.4 - Prob. 31E Ch. 13.4 - Prob. 32E Ch. 13.4 - Prob. 33E Ch. 13.4 - Prob. 34E Ch. 13.4 - Prob. 35E Ch. 13.4 - Prob. 36E Ch. 13.4 - Prob. 37E Ch. 13.4 - Prob. 38E Ch. 13.4 - Prob. 39E Ch. 13.4 - Prob. 40E Ch. 13.4 - Prob. 41E Ch. 13.4 - Prob. 42E Ch. 13.4 - Prob. 43E Ch. 13.4 - Prob. 44E Ch. 13.4 - Prob. 45E Ch. 13.4 - Prob. 46E Ch. 13.4 - Prob. 47E Ch. 13.4 - Prob. 48E Ch. 13.4 - Prob. 49E Ch. 13.4 - Prob. 50E Ch. 13.4 - Prob. 51E Ch. 13.4 - Prob. 52E Ch. 13.4 - Prob. 53E Ch. 13.4 - Prob. 54E Ch. 13.4 - Prob. 55E Ch. 13.4 - If h(x) = 2x5 + 5x4, find h(a + 3) with the aid of...Ch. 13 - In Problems 16, determine if the sequence is...Ch. 13 - Prob. 2PCT Ch. 13 - Prob. 3PCT Ch. 13 - Prob. 4PCT Ch. 13 - Prob. 5PCT Ch. 13 - Prob. 6PCT Ch. 13 - Prob. 7PCT Ch. 13 - Prob. 8PCT Ch. 13 - Prob. 9PCT Ch. 13 - Prob. 10PCT Ch. 13 - Prob. 11PCT Ch. 13 - Prob. 12PCT Ch. 13 - In problem 1315, find the indicated sum. 13....Ch. 13 - Prob. 14PCT Ch. 13 - Prob. 15PCT Ch. 13 - Prob. 16PCT Ch. 13 - Prob. 1RE Ch. 13 - Prob. 2RE Ch. 13 - Prob. 3RE Ch. 13 - Prob. 4RE Ch. 13 - Prob. 5RE Ch. 13 - Prob. 6RE Ch. 13 - Prob. 7RE Ch. 13 - Prob. 8RE Ch. 13 - Prob. 9RE Ch. 13 - Prob. 10RE Ch. 13 - Prob. 11RE Ch. 13 - Prob. 12RE Ch. 13 - Prob. 13RE Ch. 13 - Prob. 14RE Ch. 13 - Prob. 15RE Ch. 13 - Prob. 16RE Ch. 13 - Prob. 17RE Ch. 13 - Prob. 18RE Ch. 13 - Prob. 19RE Ch. 13 - Prob. 20RE Ch. 13 - Prob. 21RE Ch. 13 - Prob. 22RE Ch. 13 - Prob. 23RE Ch. 13 - Prob. 24RE Ch. 13 - Prob. 25RE Ch. 13 - Prob. 26RE Ch. 13 - Prob. 27RE Ch. 13 - Prob. 28RE Ch. 13 - Prob. 29RE Ch. 13 - Prob. 30RE Ch. 13 - Prob. 31RE Ch. 13 - Prob. 32RE Ch. 13 - Prob. 33RE Ch. 13 - Prob. 34RE Ch. 13 - Prob. 35RE Ch. 13 - Prob. 36RE Ch. 13 - Prob. 37RE Ch. 13 - Prob. 38RE Ch. 13 - Prob. 39RE Ch. 13 - Prob. 40RE Ch. 13 - Prob. 41RE Ch. 13 - Prob. 42RE Ch. 13 - Prob. 43RE Ch. 13 - Prob. 44RE Ch. 13 - Prob. 45RE Ch. 13 - Prob. 46RE Ch. 13 - Prob. 47RE Ch. 13 - Prob. 48RE Ch. 13 - Prob. 49RE Ch. 13 - Prob. 50RE Ch. 13 - Prob. 51RE Ch. 13 - Prob. 52RE Ch. 13 - Prob. 53RE Ch. 13 - Prob. 54RE Ch. 13 - Prob. 55RE Ch. 13 - In Problems 5356, find the sum of each geometric...Ch. 13 - Prob. 57RE Ch. 13 - Prob. 58RE Ch. 13 - Prob. 59RE Ch. 13 - Prob. 60RE Ch. 13 - Prob. 61RE Ch. 13 - Prob. 62RE Ch. 13 - Prob. 63RE Ch. 13 - Prob. 64RE Ch. 13 - Prob. 65RE Ch. 13 - Prob. 66RE Ch. 13 - Prob. 67RE Ch. 13 - Prob. 68RE Ch. 13 - Prob. 69RE Ch. 13 - Prob. 70RE Ch. 13 - Prob. 71RE Ch. 13 - Prob. 72RE Ch. 13 - Prob. 73RE Ch. 13 - Prob. 74RE Ch. 13 - Prob. 75RE Ch. 13 - Prob. 76RE Ch. 13 - Prob. 77RE Ch. 13 - Prob. 78RE Ch. 13 - Prob. 1T Ch. 13 - In Problems 16, determine whether the sequence is...Ch. 13 - Prob. 3T Ch. 13 - Prob. 4T Ch. 13 - Prob. 5T Ch. 13 - Prob. 6T Ch. 13 - Prob. 7T Ch. 13 - Prob. 8T Ch. 13 - Prob. 9T Ch. 13 - In Problems 912, write a formula for the nth term...Ch. 13 - Prob. 11T Ch. 13 - Prob. 12T Ch. 13 - Prob. 13T Ch. 13 - Prob. 14T Ch. 13 - Prob. 15T Ch. 13 - Prob. 16T Ch. 13 - Prob. 17T Ch. 13 - Prob. 18T Ch. 13 - Prob. 19T Ch. 13 - Prob. 20T Ch. 13 - Prob. 1CUR Ch. 13 - Prob. 2CUR Ch. 13 - Prob. 3CUR Ch. 13 - Prob. 4CUR Ch. 13 - Prob. 5CUR Ch. 13 - Prob. 6CUR Ch. 13 - Prob. 7CUR Ch. 13 - Prob. 8CUR Ch. 13 - Prob. 9CUR Ch. 13 - Prob. 10CUR Ch. 13 - Prob. 11CUR Ch. 13 - Prob. 12CUR Ch. 13 - Prob. 13CUR Ch. 13 - Prob. 14CUR Ch. 13 - Prob. 15CUR Ch. 13 - Prob. 16CUR Ch. 13 - Prob. 17CUR Ch. 13 - Prob. 18CUR Ch. 13 - Prob. 19CUR Ch. 13 - Prob. 20CUR Ch. 13 - Prob. 21CUR Ch. 13 - Prob. 22CUR Ch. 13 - Prob. 23CUR Ch. 13 - Prob. 24CUR Ch. 13 - Prob. 25CUR

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Chapter 13 Solutions