13. Prove or disprove the following by mathematical induction, where n is set of positive integers. a. E(31 – 2) = n( 3n-1) 2 b. E = n*(n+1)

Q: 1² +2²+ •+n² = n(n+1)(2n+1) 6 for n EN

A:

Q: Use mathematical Inductlon to prove that for r>0 and any positive Integer

A:

Q: 7n is a 15 n° Use mathematical induction to show that 5 3 natural number for all n E N. + +

A:

Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: 7n - 2" is divisible by 5. We will show that P(n) is true for every integer n 2 0. Show that P(0) is…

A: We have to prove that the given statement by the mathematical induction.

Q: 12 + 22 + ..+n² = n(n+1)(2n+1) 6. is true?

A: Here we have to show that which value of n to proof the mathematical induction in the first step.

Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € N.

A:

Q: 1² +2²+...+ n² = n(n+1)(2n+1) for 6

A: We are given expression

Q: use the Principle of Mathematical Induction to show that the given statement is true for all natural…

A: To Prove: 12+22+32+...+n2=nn+12n+16

Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

A:

Q: Prove the following fact by induction: For all n > 1, 4" > 3" + n2.

A: To Proof :-  The following fact by induction : For all n ≥1 ,     4n ≥ 3n  + n2

Q: 5. Use induction to prove that for every n E N we have E(k · k!) = (n + 1)! – 1 k=1

A: We want to prove ∑k=1n(k·k!) = (n+1)! - 1

Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: 1. Prove by induction that for all n € N>1 we have (a) (b) (c) 2 +3 2.3 9+ (9 × 10) + (9 × 100) ++…

A: (a) Given 2n+3<2n·3 For n=1 L.H.S=21+3=5 R.H.S=21·3=6 Thus the inequality hold for n=1 Let…

Q: (b) Prove the following by mathematical induction: n(n+ 1) (п+2) 1.2+2.3+3.4+….+n(n+1) = 3 for all…

A: By using mathematical induction we solve the result as follows:

Q: 16. Use mathematical induction to prove that, for all natural numbers n, n [n(n+1) i=1

A: Prnciple of mathrmatimat induction:Alogrithm              1. obtained  P(n) and understand its…

Q: Prove for any natural number n that 12 + 22 + 32 + ...+ n² = ÷n 1 (n + 1)(2n + 1).

A: Step 1: This can be proved by using Induction method. This method involves following steps. Prove…

Q: Prove the following by mathematical induction: п(n+ 1)(п+ 2) 1.2+2.3+3.4+……+n(n+1) = 3 for all…

A: We have to prove the given summation result using the principle of mathematical induction.

Q: 3. Use Mathematical Induction to prove that 4 + 10 + 18 + ... + n (n + 3) = [n .. (n + 1) (n +5)] /3…

A: The solution is given as

Q: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

A:

Q: 7. Prove by induction that E,(-1)² = (-1)"n(n + 1)/2 for all n > 1.

A:

Q: . Prove, using mathematical induction, that 13n – 8n is divisible by 5, for n ≥ 1.

A: Let us consider the statement is  P(n) : 13n - 8n is divisible by 5. For n = 1, 13n - 8n = 5 is…

Q: Prove by induction that 3" <n! for all integers n with n ≥ 7.

A: As per the answering guidelines we will solve the first question only. Please upload the other…

Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

A:

Q: 1. Prove by mathematical induction that n(n+1) (a) 1+ 2+ + 12 = 2 (b) 1+3+5++ (2n-1) = n²…

A:

Q: 2. Prove by induction that 3² i=0 3n+1-1 2 for all n with n ≥ 0.

A:

Q: 4. Prove the following statement by mathematical induction: 1 1 + 3-4 + 2.3 1 1.2 + + 1 n(n+1) - n…

A: Mathematical induction is a method for proving that a statement P(n) is true for every natural…

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: 11. Use Math Induction to prove the following formula for positive values of n: 1 A i(i+1) n+1" i=1

A: Given, ∑i=1n1ii+1=1-1n+1 Consider, Pn≡∑i=1n1ii+1=1-1n+1        ...1

Q: b. Use the Mathematical Induction to show that the given statement is true for all n ≥ Z P(n):…

A: