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First check that this works for n=1:2^(2x1 - 1) + 3^(2x1 - 1) = 2^1 +3^1 = 5 (so true for n=1)Now we assume this to work for any n = k.Assumption: 2^(2k-1) + 3^(2k-1) = 5a, where a is some integer constan...

Consider the general equationdy/dx + Py = Qwhere P and Q are functions of x.R (which will be introduced later) is also a function of x.
So, all of a sudden, we are going to just state the product rul...

As proof by induction goes we always have to show that it works for the base case, which in this case is the very first positive integer:1. So we show that the sum of the first number(s) is equal to one w...

3sinh^2(2x) + 11sinh(2x) - 4 = 0 --> (3sinh(2x) - 1)(sinh(2x) + 4) = 0 --> sinh(2x) = 1/3, sinh(2x) = -4(e^(2x) - e^(-2x))/2 = 1/3 --> e^(4x) -(2/3)e^(2x) - 1 = 0 --> e^(2x) = 1/3 + 2sqrt(5)/3...

(1+2i) (2+i)/(2+i)(2-i) = (2-2+4i+i)/5 = 5i/5 = i