1.21: Past-paper question 7

(a) Use the factor theorem to show that $\displaystyle (4x - 1)$ is a factor of

(b) Hence, or otherwise, find the exact roots of the equation

A geometric series $\displaystyle G$ has first term $\displaystyle a$ and common ratio $\displaystyle r$
The third term of $\displaystyle G$ is 9 and the sum to infinity of $\displaystyle G$ is 192

(c) Show that $\displaystyle 64r^3 - 64r^2 + 3 = 0$

Given that $\displaystyle r$ is a rational number

(d) write down the value of $\displaystyle r$

(e) show that $\displaystyle a = 144$

The sum to $\displaystyle n$ terms of $\displaystyle G$ is $\displaystyle S_n$

(f) Using logarithms, find the least value of $\displaystyle n$ such that $\displaystyle S_n > 191.9$