In Exercises 1-4, use the Principle of Induction to prove the formula for all natural numbers n. n(n + 1) 1. 1+2+3+ ...+n = 2 n²(n + 1)2 2. 13 + 23 + 33 + ·. +n3 4 1 3. 1. 2 1 +... + 1 п(п + 1) n + 1 1- x"+1 1- x 4. 1+x+ x² +...+x" = for any x + 1

Q: 10. Using the first principle of math induction, prove that, for n positive integer n³ + 2n is…

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Q: 6) Apply induction to show that 2+4+ .2n n(n+1), n 21.

A: We prove this by the induction, firstly prove for n=1, then assume it is true for n=k, and then…

Q: Using mathematical induction method prove that for every positive integer n, 5 n° – n.

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Q: 1. Use mathematical induction to prove а) 1 - 1! + 2- 2! + ... + n:n! 3D (п +1)! — 1 for all n 2 1.

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Q: Show by induction that 1+3 +5+ . + (2n – 1) = n² ...

A: See the attachment

Q: |(1!)+3(2!)+3(5!) +... + n(n!) = (n+1)!-1 for all n2 I

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Q: Use mathematical induction to prove that the sum of the squares of the first n natural numbers : Ek?…

A: Let's prove.

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

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Q: Use Principle of Mathematical Induction to show that for all n E N, an = 2n+2. 52n+1 + 3"+2. 22n+1…

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Q: Using mathematical induction, prove P(n) = 22n – 1 is divisible by 3 for every integer n > 1. Show…

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Q: Use mathematical induction to prove: 3 + 8 + 15 + 24 + ……….+ (n2+2n) = (1/6) n(n+1) (2n + 7) for all…

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Q: Use mathematical induction to prove that, for all integers ? ≥ 1, 12 + 22 + 32... + n2 = 1/6n (n +…

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Q: Use mathematical induction to prove that 3"7" - 2 is divisible by 8 for every non-negative integer…

A: Initial case:

Q: Show using mathematical induction that, for all n ≥ 1, 1 · 2 + 2 · 3 + 3 · 4 + · · · + n(n + 1) =…

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Q: Use mathematical induction to prove the formula for every positive integer n.

A: Let the given statement be PxThen px: 1+2+3+.....+n =nn+12, for  every positiveinteger nNow , P1 :…

Q: Using Induction …. Show that if n is a positive integer, then   1 + 3 + 5 + …. + (2n-1) = n2

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Q: Prove by Mathematical Induction: VneZ+, 1 · 2 · 3+2·3·4++ n · (n + 1) · (n + 2) = = n(n+1)(n+2)(n+3)…

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Q: Use mathematical induction to prove that 12 + 32 + 52 + .…+ (2n + 1)² = (n+1)(2n+1)(2n+3) where is a…

A: We need to prove the given result by mathematical induction. The solution is as follows.