Consider the numbers sn defined by sn = 1 + 3 + 5 + 7 + … + (2 * n + 1), for all n ∈ ℕ. That is, sn  is the sum of the first n + 1 odd numbers.

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

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Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: Verify that the following inequality is true for all integers n > 1 1 1.3.5. .... (2n 1) - 2n…

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Q: Find the sum 1 C|~ SA 3 π² 4² (21) + 44 (41) 4 (61) +

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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Q: 100 Σ (3* + 2k - 1) k=1

A: ∑k=11003k+2k-1

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

A: We are given expression

Q: For ne N, 2n -8 < n²-8n + 17.

A: 3.7 We have to prove that 2n−8&lt;n2−8n+17 for all n∈N by principle of induction method. Let n=1…

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

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Q: Prove that the sum 2 + 22 + 23 + 24 + · . . + 22019 + 22020 is divisible by 30.

A: The sum of the geometric series a, ar, ar2, ar3,…, arn-1 is given as the following formula.…

Q: ) Prove every integer n >= 7, n! > 3n

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Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

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Q: Prove the following statement by mathematical induction n² 2 2n for all n > 1

A: we have to solve given problem:

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

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Q: Find the sum of 1/7 + 3/7 + 32/7 + 32 /7 + ⋅⋅⋅ + (3)n-1 / 7

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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 any methods of proof to verify that n 1² + 2² + 3² + ... + n² = 1² (n · (n + 1)(2n + 1) 6

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Q: Show that yn for all n=1, 2₁- ✓

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Q: Choose all those that represent the sum 5+7+9+11+13+...+101. 101 Σκ k = 5 50 E (2K+ 1) j = 2 50 Σ…

A: We have to find the correct representation of the sum 5+7+9+11+13+⋯+101

Q: 8] n=1 1 5n2 + 5n

A: To Find: The sum ∑n=1∞15n2+5n

Q: Prove or disprove "Let n be the positive integer. Then, 2n + 1 ≤ 2n+1 - 2n-1."

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Q: n5 7n Prove that 3 is an integer for n > 1. 15

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Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: um 1+4+7+..

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Q: What is the sum of 1/4 + 2/4 + 3/4 + 1 + 1 1/4 +… + 9 1/4 + 9 2/4 + 9 3/4 + 10?

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Q: Evaluate the sum 50 Σ(2 · k + 5) = k=1 50 Σ5₁(2. k+5). k=1

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