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Chapter 4: Q. 61 (page 327)

Prove part (b) of theorem 4.4 in the case when n is even: if n is a positive even integer, then鈭�k=1nk=n(n+1)2

Short Answer

Expert verified

We proved鈭�k=1nk=n(n+1)2

Step by step solution

01

Given information

We are given a relation we have to prove it

鈭�k=1nk=n(n+1)2, Where n is even

02

Prove using mathematical induction

First check whether the above relation is true for n=2

We have,

LHS=3

And

RHS=n(n+1)2RHS=2(2+1)2RHS=3

Hence LHS=RHS

Now consider that the above statement is true for n where n is an even number

We have

鈭�k=1nk=n(n+1)2

Now we prove that the statement is true for n+2

That is

鈭�k=1n+2k=(n+2)(n+3)2

Consider LHS

role="math" localid="1648555801521" 鈭�k=1n+2k=鈭�k=1nk+n+1+n+2=n(n+1)2+2n+3=n(n+1)+4n+62=n2+n+5n+62=(n+2)(n+3)2

Hence proved

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Most popular questions from this chapter

Given a simple proof that鈭�k=5n(ak+bk)=鈭�k=5nak+鈭�k=5nbk

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�鈭�32鈥�1x+5dx

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to|鈭�abf(x)dx|.

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = -x2on the interval [鈭�3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by 鈭�ab(f(x)-g(x))dx.

(e) True or False: The average value of the function f(x) = x2-3 on [2, 6] is

f(6)+f(2)2= 33+12= 17.

(f) True or False: The average value of the function f(x) = x2-3on [2, 6] is f(6)-f(2)4= 33-14= 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value.

limn鈫�鈭�鈭�k=1nk2+k+1n

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�13(x+1)2dx
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