Let C(n) be the constant term in the expansion of (x + 3)". Prove by induction that C(n) = 3 for all n E N. (Induction on n.) The constant term of (x + 3)¹ is 3 Suppose as inductive hypothesis that the constant term of (x + 3)k-1 is Then (x + 3) = (x + 3)k-1. so its constant term is for some k > 1. . 3 = , as required.

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: =-rگت( (1+ V5) )(n- 2) < fn (п— 2) Use induction to prove that 2 ((1+ |(n–1). for all n 2 3.

A: Introduction: One of most important sequences in mathematics and number theory is Fibonacci…

Q: Prove by mathematical induction that 1^3 + 2^3 + 3^3 + · · · + n^3 = (n(n + 1)/2)^2 for all n ≥ 1.

A: We have to prove the given statement by mathematical induction.Base Case:Let's start by verifying…

Q: Find the coefficient of a-2 in the binomial expansion of  (27a2 - (1/81a3))9

A: Let us consider the expansion x+yn The binomial expansion is obtained as x+yn=∑k=0nCknxn-kyn

Q: (7) Use congruences to prove the following. They were proven by induction in Prob- lem 19 in Section…

A: (a) To prove that 9 divides (4^n+15n-1), we need to show that (4^n+15n-1) is congruent to 0 mod 9.…

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: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: Let cn = 4Cn-1-4-2 with co = -3 and c₁ = 4. Prove that ê= -3(2)" +5n(2)" solves the recurrence for…

A: The given recurrence relation is, cn=4cn-1-4cn-2. Condition is c0=-3 and c1=4. We have to prove…

Q: Consider a population that grows according to the recursive rule P = Pn-1 + 110, with initial…

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Q: Given the recurrence T(n)= T(n-1) +6n-5; T(0) = 4 Use mathematical induction to prove that = 3n²-2n…

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Q: Consider the recurrence t(n) = 8t(n-1) + 3n n 1.One of the roots of the characteristic polynomial…

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Q: Q4. Use Strong induction to show that Let an be the sequenced defined by 90=0 7 2Step: Case: a =…

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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: (2) (5 points each) Use the principal of induction to show that for all k = N: (a) 13 +23+33 + + k³…

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Q: Problem 8 (Mathematical Induction – : Let P(n) be the statement: "1/(1 × 3) + 1/(3 × 5) + ... +…

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Q: 3. Use the binomial theorem to prove that 2n 2n |22n–2k = (2.5)2". k k=0

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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