PLEASE TYPE ONLY***EXERCISE 4.11.2 ONLY DO (E-F)**** Exercise 4.11.2: Proving identities by induction.Prove each of the following statements using mathematical induction.(a)п(п + 1)Prove that for any positive integer n,2j=1(b)Prove that for any positive integer n, j · 2(n – 1)2"+1 + 2-j=1(c)Prove that for any positive integer n, ili – 1) = "(n² – 1)j=1-(d)Prove that for any positive integer n,11j(j+1)1-n +1(e)Prove that for any non-negative integer n, 33In 3"+1 - (n + 1)3" + 1]j=0(f)Prove that for any positive integer n2 2,1n +1122321n22n

Answered: (e) Prove that for any non-negative…

(e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for any positive integer n2 2, n +1 1 22 1 n2 - | - ... 32 2n

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 32E: Use the second principle of Finite Induction to prove that every positive integer n can be expressed...

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

PLEASE TYPE ONLY***

EXERCISE 4.11.2 ONLY DO (E-F)****

Exercise 4.11.2: Proving identities by induction.
Prove each of the following statements using mathematical induction.
(a)
п(п + 1)
Prove that for any positive integer n,
2
j=1
(b)
Prove that for any positive integer n, j · 2
(n – 1)2"+1 + 2
-
j=1
(c)
Prove that for any positive integer n, ili – 1) = "(n² – 1)
j=1
-
(d)
Prove that for any positive integer n,
1
1
j(j+1)
1-
n +1
(e)
Prove that for any non-negative integer n, 3
3
In 3"+1 - (n + 1)3" + 1]
j=0
(f)
Prove that for any positive integer n2 2,
1
n +1
1
22
32
1
n2
2n

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