1. Use induction to prove that, for n>=0: 3*50 +3*5¹ +352 +3*5³+...+ 3*5=3*(5+¹-1)/4 In other words: Basis: Assume: Prove: n Σ k= 0 3*5 k ==3*(5+1-1) / 4

Q: Show that the middle term in the expansion of (1 + x)2¹ is (1-3.5 (2n- -1)) 2"x", where n is a…

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Q: Show that for all n> 14, n cents can be represented by some combination of 3-cent coins and 8-cent…

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Q: Q3 Use Principle of Mathematical Induction to show that for all n EN, an = 2"+2. 52n+1 + 3n+2 .…

A: As per our guidelines we can solve only one question at a time. Please repost unsolved questions…

Q: n(n + 1)(2n + 1) 2. Use induction to prove that )k2 for all n e Z+. %3D 6. k=1

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Q: Let n E N. Prove by mathematical induction that 3* = 4". k k=0

A: We have to prove given expression using mathematical induction.

Q: =. Prove by mathematical induction: n-1 Σία + 1 = i=1 For every integer n ≥ 2. n(n – 1)(n + 1) 3

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Q: Consider the sum (3)² + (8)² + (13)2 + · . + (15n – 2)2. | m 1 1 m(т + 1)] and k2 = (m(m + 1)(2m +…

A: Consider S=32+82+132+ . . . .+15n-22

Q: Σ1 (−1) = (n+1)(-1)", 2

A: We have to prove  Prove that ∑ni=1 i2(−1)i = ((n+1)(−1)nn)/2 , where n is a positive integer. We…

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

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Q: 2. Using induction, prove that 1 %3D i(i+1) n+1

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Q: 12 Let u₁ = 1, ₂ = 1 and un + 2 = Un + 1 + un n ≥ 1. Then use mathematical induction to show that 1…

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Q: Prove using mathematical induction that for all n E N. i E7-1(8i – 5) = 4n2 – n %3D ii. 52n - 1 is…

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

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Q: e 27 > 3n2 +3n +1 for all n E Z†, n > 8

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Q: 1. Prove that for all positive integers n, n Σj2i = (n − 1)2n+¹ +2 j=1

A: To prove that,              ∑j=1nj2j=n-12n+1+2.

Q: Show that n is composite if and only if p(n) <n – Vn.

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Q: 6. The Ackermann's function is given: if m = 0 if m > 1 and n = 0 if m 2 1 and n = 1 la(m – 1,A(m,n…

A: Ackermann's function is defined as Am,n=2n ,                            m=0=0 ,…

Q: 10.) For all n > 1, prove the following by mathematical induction: 1 +. 1 1 1 (а) 12 <2 - n2 .. 22…

A: solutionLet p(n):12+222+322+......+n2n=2-n+22n    ∀ n &gt;,1

Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: -1/2 p. Find the coefficient of x" in the expansion of (1– 4x)2.

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Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: n (3n–1) 2 B. Use mathematical induction to prove that P(n): 1+4+7+..+(3n-2) = n2 1. Use the first 2…

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Q: Prove by induction: 1) 2 +4 + 6 + 2) 1 +4 +7+ + 2n = n (n + 1) .. + (3n – 2) = 1/2 n ( 3n - 1)

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Q: Prove:  C(n,k) = C(n-1,k-1)+C(n-1,k) for natural numbers k with 0<k<n.

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Q: a) Prove that ε n(n+ 1)(2n + 1) Σ 6 i=0

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