1=1,f2 =1 and fn+2 =fn +fn+1 for all n≥1.    Prove that for all natural numbers n≥1: f1 f2 +f2 f3 +f3 f4 +...+f2n-1 f2n =f

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: N.Find the set of values of x for which: w - x-2 x+1 12. Solve the inequality:(x-3)(x+2)(x+4) > C 13…

A: 11)  xx-2>1x+1⇒xx-2-1x+1>0⇒xx+1-x+2x-2x+1>0⇒x2-2x-2x+1>0           1 If x<-1 then…

Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

A:

Q: Let S(n) denote the statement n(n + 1) + (n - 1)(п - 2)(n - 3) (n 4) (n -5) (n - 6) (п -7). 2 his n…

A: (a)

Q: Let f(x) = Ax² + Bx + C where A, B and C are real numbers. Prove that if f(x) is an integer whenever…

A:

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: Please don't provide handwritten solution ..

A: The symbol Γ(n) represents the gamma function, which is an extension of the factorial function to…

Q: how that 1+2+2²+23+...+2 = 2n+1 -1 for n≥ 0.

A: We need to prove by induction hypothesis      1+2+22+.........+2n=2n+1-1, for n≥0

Q: Let b a natural number and q1, q2, r1, r2 integers with 0 ≤ r1 < b and 0 ≤ r2 < b such that q1b + r1…

A: Natural Numbers: Natural numbers, that embody all positive integers from one to time, square measure…

Q: this

A: We use definition of t(n).

Q: [16] 1. Represent each pair of sets A and B on a number line, then find AUB and An B in interval…

A:

Q: Find a one-to-one correspondence between [0, 1] and (0, 1). Find a one-to-one correspondence between…

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Q: Prove or disprove: There exists an integer a such that 14 | (3a – 5) and 21 | (2a +3).

A:

Q: Let r and s be nonnegative real numbers, f(r) = r" and g(r) = r. %3D Find all pairs r and s such…

A: Given that: f(x)=xr           and             g(x)=xs

Q: Va +3b + Va – 3b If x=- Va+36 – Va– 3b prove that 3bx² – 2ax +3b = 0 | - -

A:

Q: (a) Let n be a non-negative integer. Prove that if n is a multiple of 3, then n² is a multiple of 3.…

A: (a) Let n be a non-negative integer and be a multiple of 3.       Then, there exists some…

Q: The number of X₁ + X2 + x3 + x4 + X5 + X6 = 12 is C(12+6-1, 12) = C(17, 12). True non-negative…

A: It is known that the number of non-negative integer solutions to the equation: x1+x2+x3+⋯+xn=k is…

Q: Find L S s²-48-7 Find L¹ 2s+6 s² +6s+34

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Q: k + 6 k+6 Hint: 126 = -te 3 Assume that n is a positive integer. Compute k+1 k3 + 3k² + 2k* k=1

A:

Q: (b) 6 + 7 + 8 + 9 + ... + k

A: (b) Given that 6+7+8+9+...+k The objective is to find the sum. The formula for sum of numbers are,…

Q: Prove by, Proof by Contradiction.   (a) If n is an integer, then n2+ n3 is an even number.   (b) If…

A:

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.