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Prove by induction that f(n) = 2^(k + 2) + 3^(3k + 1) is divisible by 7 for all positive n.

First we establish our base case: f(0) = 2 2 + 3 1 = 4 + 3 = 7, so clearly f(0) is divisible by 7.Now. by the inductive hypothesis. we assume that f(k) is divisible by 7, and attempt to show that this implie...
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How do you define a class in python

Here's a simple example of a class:class Animal: def init (self, name, noise): self.name = name self.noise = noise def make_noise(self): print(self.noise) here init is the "initializer" (or sometim...
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Answered by William P. • Python tutor
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how do I solve the equation x^2 + 7x + 11 = 0, to 3dp?

This is a quadratic equation and there are three ways to solve one. This question specifies that the answer should be left to three decimal places. This tells you that you can not factorise, as your answer w...
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Answered by Jamie D. • Maths tutor
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Let f (x) = 5x and g(x) = x2 + 1 , for x ∈  . (a) Find f-1(x) . (b) Find ( f ° g) (7) .

a,y=5xfinding the inverse: x=5y - dividing by 5 (1/5)x=y f -1 (x)=(1/5)x b,5x(7 2 +1)=250
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Differentiate f = ln(x^2 + 1) / (x ^ 2 + 1).

We'll apply the quotient rule using "u = ln(x 2 + 1)" and "v = x 2 + 1". First we'll need to calculate u' and v'.Using normal differentiation rules, we can see "v' = 2x". Now th...
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Answered by William P. • Maths tutor
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