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

Q: Prove by induction that 1.3.5 (2n-1) 2.4.6(2n) for n a positive integer. VI +1

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Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

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: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: Show that for positive integer n,

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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: 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: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

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Q: 8. Use mathematical induction to prove "for every positive integer n, n³ - n is divisible by 3".

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Q: (Prove by induction that for all integers n ≥ 1, 1² + 32 +5² +...+(2n-1)² = n(2n-1)(2n + 1) 3

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Q: . By mathematical induction, prove that if ニ then A" =| 1 + 10n 4n -25 -25n 1- 10n 11 A - 9

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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: Prove that 1 + 2 + 22 + 23 + . + 2 2n-1 1 for n2 1. Use proof by induction.

A: Given  1+2+22+23 + . . . + 2n = 2n+1-1 for n≥1.

Q: Use induction to prove that 3 | (10n+1 + 10n + 1) for all positive integers n

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

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Q: SOLVE STEP BY STEP IN DIGITAL FORMAT Prove by induction that 1² +2² +3²+...+ n² n(n + 1)(2n + 1) 6

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

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

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

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Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: Use mathematical induction to prove the following: (a) 1×3+2×4+3×5+...+n×(n+2)= "x(n+D(2n+7); for…

A: as per our guidelines we can provide answer to one question at a time you can repost others

Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

Q: Use the principle of m matical induction to prove that for every positive integer n, the quantity…

A: Principle of mathematical induction: This method is used to prove a statement or theorem. In this,…

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

A: As per the question we have to use induction to prove that :12 + 22 + ... + n2 = n(n+1)(2n+1)/6

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.