Alan

As follows:

This graph might help:

The common ratio here is r = -(2/3)², so the infinite sum is given by (3/2)/(1 - r) = 27/26 ≈ 1.038

As follows:

9876ⁿ (mod(100)  = 9876 (mod(100)) = 76

Hence  $\Sigma_{n=1}^{99}9876^n (\mod(100) ) = 99*76 = 7524$

Let the distance from A to B be L miles

time from A to B:  t1 = L/40 hours

time from B to A:  t2 = L/60 hours

Average speed for round trip = total distance/total time

so average speed = 2L/(t1 + t2)  = 2L/(L/40 + L/60) = 2/(1/40 + 1/60)  = 48 mph

No, you coud have x = 6/y as well.

You could also draw a graph, which often helps in these sort of questions:

It isn't!

Let n = 1

20² + 16² - 3 = 653

653/323 = 2.02167...

I think you must mean prove that   $20^{2n}+16^{2n}-3^{2n}-1$  is divisible by 323 for integer n

Find a function f(x) such that f'(x)=8x^2+5  and f (0) =1

You are looking for the antiderivative of f'(x)

The antiderivative of x²  is  x³/3,  that of x⁰ is x¹/1  (Note that x⁰ is just 1 and x¹/1 is just x).  

So f(x) = 8x³/3 + 5x + k  where k is a constant to be found by using the fact that f(0) = 1.

As follows