Let C(n) be the constant term in the expansion of (x+5)". Prove by induction that C(n) = 5", for all n E N.

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: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

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Q: Show that 1+2+3+… +n = (";").

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Q: 11. Prove by mathematical induction that > k (k – 1) = (n + 1)n(n – 1) 1sksn

A: To prove:∑1≤k≤nk(k−1)=13(n+1)(n)(n−1)Proposition:- For any n∈N∑1⩽k≤nk(k−1)=13(n+1)n(n−1)⇒ 1(1-1) +…

Q: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

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Q: Let C(n) be the constant term in the expansion of (x + 7)". Prove by induction that C(n) = 7" for…

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Q: n-1 For all n > 2, 4+ 3n4" – 4n+1 - j=1 9.

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Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: use mathematical induction to prove, for n > 1: 4n3 E(2k – 1)? = | 3 k=1

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Q: Use Mathematical Induction to show that, for all n € Z†, 6 | (10″ + 2).

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Consider the following recursion (suppose n is a power of 2): √4T(n/2) +n².5 if n > 1 if n = 1 T(n):…

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Q: . Use Mathematical Induction to show that for all n > 1. 71 i=1 F2i = F2n11-1

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Q: 15. a. Prove the following identity using induction on k. n + n+ 2 n (") + (˜† ¹ ) + (˜ + ²) + ··· +…

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Q: Prove by induction that E-,(i × 2' +1) = 4(n × 2" – 2" +1). 'i = 1

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Q: Use the Binomial Theorem to find the coefficient of x' in the expansion of (x+3)" ...-.

A: on using binomial theorem we get the coefficient of x^7 that is

Q: Give a recursive definition (with initial condition(s)) of (an) where (n = 1, 2, 3, . . . )

A: We have, an = 212nWe have to find a recursive definition with initial conditions of an starting from…

Q: a(1–r") for every 1-r Let r + 1 be a real number. Use induction to prove that a + ar + ar² + . ..+…

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Q: An = 2n an = 3n – 5 an = (n + 1)/3 an = v2

A: Given:  a) an=2n b) an=3n-5 c) an=n+13 d) an=2

Q: The sequence (sn) is defined recursively by s­­­0=3 and sk=sk-1+2k. Show that sn=3+n(n+1) using…

A: We are going use method of mathematical induction to prove the result

Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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