problem 1: The eighth term of an arithmetic progression is twice the fourth term. The seventh term is 14. Find the sum of 100 terms.    

problem 2: The fourth term of a geometric progression is 4, and the seventh term is 0.5. Find the sum to infinity for this geometric progression.

problem 3: A geometric progression has the first term 10 and common ratio 1.5. The sum to n terms is 131.875. Find the value of n.    

problem 4:

a) Differentiate the following functions:

f(x) = 1/x
g(x) = √(x²)³

b) Differentiate the following functions:        

f(z) = z³ (z^-3/4 + z^3/4 + z)

g(z) = 4/z³ – z^3/4

problem 5: Differentiate the following functions with respect to x:        

a) f(x) = (x² + 2x) exp(x)

b) g(x) = 1/(x² – 1)

c) h(x) = exp(x(x+1))

d) j(x) = ln(1 + x + x²)

e) k(x) = e^x/(e^x – e^-x)

problem 6: You are given the function f(x) = x² - 3x + 2.

a) Find the equations of the tangents at the points where the function crosses the x-axis.
b) Find the point where the two tangents intersect.