15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

Q: 15. Prove that for each natural number n, 5+ ...+ (4n-3)(4n+1) 4n+1'

A:

Q: Prove 22n! for n=4 Prove R

A: According  to guidelines  we solve one question. We solve question 1 by induction method.

Q: 1. If n is an integer, then 2n +1 is odd.

A:

Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

Q: 1 Prove that + n is an integer for any n E N. 6

A:

Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: Prove or disprove: for every natural number n,2+5+8+...+(3n-1)=n(3n+1)/2

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Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

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Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: For each natural number n, let 1 n+1) = {r €R: <r< 2n+1} . An n2 Calculate (with proof) U An and N…

A: The objective is to calculate:

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

A:

Q: Prove there are no natural numbers n and p with n 2 2 and p² + 1 = 2".

A: prove by contradiction:- to prove it by contradiction, you should start by assuming its negation,…

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

A:

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Prove n (2n = n n 1 k=1

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Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

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Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

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Q: 1. Prove by induction that for all positive integers n, we have: 1+3+5+ + (2n 1) n2

A:

Q: 1. Prove: If n is odd then 5n + 4 is odd.

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Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: 1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

A:

Q: 8. (Chapter 10, #16) Prove that for every positive integer n, 2" +1< 3".

A: Using induction to prove any statement includes first proving the statement is true for n=1, then…

Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

Q: 1. Use mathematical induction to show that n2 – 3n +4 is even for all n > 1.

A: To show that n2-3n+4 is even for all n≥1, using Mathematical induction.

Q: 4. Prove that for every n > 1,3 divides n³ – n.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity involving the sum to n terms of the given series

Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

A:

Q: Prove by induction: 1 1+i+5+* for all n > 1.

A:

Q: 4. Let bo = 1, bị = 3, and for each integer k > 2, we have bk = 4bk–1 – 4bk–2 Prove that for all…

A:

Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity giving the sum of the series

Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.