Let F, denote the nth Fibonacci number (F1 = F2 = 1, Fn+2 Fn+1 + Fn for n > 1). Use %3D induction to prove that Vn > 1: Fn - Fn+1Fn – F = (-1)"

Q: Give an inductive proof of the following identity where Fn is the math Fibonacci number F0^2 + F1^2…

A: To prove  f02+f12+f22+...+fn2=fnfn+1          1 where fn is the Fibonacci number

Q: Prove by induction E 3₁ = (3^+¹ -1) 2 i=o

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Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: Show that the identity k n = n 72 n k k-1, olds for all integers 1 <k<n and use it to prove that n…

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Q: Part III: Prove the following 5. Prove that 32.753215321... is a rational number.

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Q: Let yk = k", where n is a positive integer; then Ak" = (k + 1)" – k" %3D (1.112) n = nk"-1 kn~2…

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Q: Prove that Q_n = sup 1/n(1-(1/(nx+1))=1/n

A: Answer

Q: List all numbers x such that 0≤x<100 and x ≡ [3,1] mod [4,5]

A: Given,           0≤x&lt;100, and  x ≡ [3,1] mod [4,5].

Q: Sp

A: The objective of this question is to understand the formula for 'n choose k', which is a concept in…

Q: show that 2 is a primitive root mod 3^e for all e>=1

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Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

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Q: Use Euler's Theorem to find a number x, 0 <x<7, such that x^21 = 3 (mod 8)

A: By Euler's theorem, for any number x such that 0 &lt; x ≤ 7,        xφ(8) ≡ 1 (mod 8)       =&gt; x4…

Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: How many ways are there to distribute r+s identical balls among five different children such that…

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Q: m+1 ) w = n+1 [ + w U w + m-1 n-1 n

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Q: 15. Use a 30-60-90 triangle to find the tangent of 60°. 2 /60° 1 30% √3

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Q: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

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Q: Let p=17. If a belongs to F_{17} and a is non zero , compute the value of a^{16}(mod 17) for a=3 .

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Q: ind e so that 23^e = 1(mod 21)

A: Here we use Euler’s phi theorem: Let n be a positive integer and a an integer relatively prime with…