. Use mathematical induction to prove each of the following: * (a) For each natural number n, 2+5+8+...+(3n − 1) = = (b) For each natural number n, 1 +5+9+ + (4n − 3) = n (2n − 1). n (3n + 1) 2 n (n+1)]² 2 (c) For each natural number n, 1³ +2³ +33 +...+n n³ = .

Q: In each of the following statements, n represents a positive integer. One of the statements is true,…

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Q: For a natural number n, we define n! = 1·2·3· ... · (n − 1) · n. For example: 2! = 1 2,3! = 1 2 3 =…

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

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

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Q: Ci) Use the method of Mathematical Induction to show that for all positive integer values n21,…

A: The principle of mathematical induction is a technique of proving a statement or formula which is…

Q: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

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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: For every positive integer n, 1 + 1.4 4.7 + ... + 1 (3n - 2)(3n+1) n = 3n+1

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Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

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Q: 1.-Proof by mathematical induction that 5 + 9 + 13 + ··· + (4n + 1) = n(2n + 3) ...

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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: (a) Which primes can divide n² + 1 for some n? a 30000 40 AT & opinip ad seppo 90.4/3 (4) рро t' for…

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Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: For each positive number n, let P(n)be the formula 1 1 (1+;)(1+ (1+– = n + 1. .. (a) Write P(1). Is…

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Q: (1) Use induction to prove the following for all natural numbers n: 13 + 23 +... n3 n² (n+1)² 4

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Q: induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: To prove by induction :Let the proposition P(n): 12 + 22 + ... + n2 = n(n+1)(2n+1)/6Here n is…

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

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Q: Use induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6 for all n>0.

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