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Prove the basis to be true. Let n=1 and this gives f(1)=16+375=391 which is divisible by 17. Now assume that if we let n=k f(k) is divisible by 17. If we now let n=k+1 and prove f(k+1) is divisible by 17 ...

Solving the ansatz equation x^2 - 4x + 3 = gives 2 equal roots where x = 3 and x = 1The general solution therefore is y = Ae^3x + Be^x where A and B are arbitrary constants

If you have studied Physics, you may be familiar with the small angle approximation cos θ ≈ 1 - θ²  / 2. This is a surprisingly accurate approximation for small values of theta (try it!), but w...

by completing the square we write the equation as (x+b/2)^2-b/2^2+c, in this case b=8 (the coefficient of x) and c=5 so we have (x+4)^2-16-5=0, which equals (x+4)^2-21=0. Now by rearranging we get (x+4)^2...

Here we can use integration by parts. Notice that ln(x) can be written as ln(x)1, so we can integrate 1 and differentiate ln(x).
Then using the formula int(uv') dx = uv - int(u'v) dx, we...