1. Use induction to prove that 12-22 +32...+-1)-¹n² = (-1)n-1 n(n+1) for all positive integers n. 2

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

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Q: For any positive integer n, n(n+1)(2n+ 1) EP = 12 +22 + .. + r?

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Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: Use mathematical induction to prove the following claim for all integers n > (2i - 3) = (n² – 2n +…

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Q: n (n+1)(n+ 1) Eli+1) = 2 j=0

A: Principal of mathematical Induction

Q: Use a mathematical induction to prove that (n(n+1) Sn: 13 + 23 + .. + n³ = 2 is true for all…

A: Sn : 13+23+....+n3=nn+122 When n=1, The value of left hand side is, 13=1 The value of right hand…

Q: Use induction to prove that 5|(8" – 3") for all n E N.

A: Use induction to prove that 5/8n-3n for all n∈ℕ.

Q: Prove by the method of induction for n >1: 1, 1 1 1 +- 1-3 3.5 5-7 (2n-1)(2n+1) 2n+1

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

A: We are given expression

Q: Using mathematical induction, prove that for all nonnegative integers n 2n+1 + (-1)" 3. 2" %3D j=0

A: For the solution of the problem follow the next steps.

Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: 4. Prove by induction that 3" ≥ 2" +n for all n ≥ 1.

A: We have to show that assuming the statement is true for n = k we have to prove the statement is true…

Q: (1) Prove by simple induction that, for each integer n 2 1, 6+62 + 63 + .. +6" 6+1 - %3D

A: Introduction: Mathematical induction is a mathematical technique for proving that a statement,…

Q: c. Prove using weak induction that for all positive integers n 1² + 3² +5² +...+(2n − 1)² : - = 4n³…

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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: (b) Prove by induction for all n >1 that an – bn = 2bn-1· %3D

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Q: Prove (using induction) that for all integers n ≥ 1 1 +6+11+ 16 + + (5n-4) · ... = n(5n - 3) 2

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Q: # 2. Use mathematical induction II to prove that for all integers n 2 5, 1 + 4n < 2".

A: Given: n≥5 To prove: 1+4n&lt;2n using mathematical induction.

Q: 3. Prove each of these divisibility statements by mathematical induction. (b) (a) 8|52n +7 for all n…

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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: Check, Is function x4 g(x, y) = =√x² + y² 0 If ContinOus at (0,0)? If (x, y) = (0,0) (x,y) = (0,0)

A: The given problem is to check whether the given function g(x,y) is continous at (0,0).

Q: . Use mathematical induction to prove each of the following: * (a) For each natural number n,…

A: (a). To prove that for each natural number n,                         2+5+8+. . . +(3n-1)=n(3n+1)2.