In an arithmetic sequence, the first term is 2, and the fourth term is 14.

a) Find the common difference, d.

b) Calculate the sum of the first 14 terms, $latex S_{14}$.

Let $latex log_{b}\, 2=u$, and $latex log_{b}\, 5=v$. Find an expression in terms of u and v for:

a) $latex log_{b}\, 10$.

b) $latex log_{b}\, 25$.

Find the $latex x^{3}$ term in the expansion of $latex \left ( x-\frac{2}{x} \right )^{5}$.

The first 3 terms in a geometric sequence are: $latex k^{2}, \: -k, \: k-2$.

a) Find the value of k.

b) Find the sum of the terms to infinity.

In an arithmetic series, $latex S_{20}=670$, and $latex u_{20}=62$.

a) What is the first term, $latex u_{1}$?

b) What is the common difference, d?

Solve the following sum:

$latex \sum_{n=1}^{100}2n+3$

Simplify $latex \frac{\left (-6 x^{-2}y^{2} \right ) \left (-8x^{-5}y^{-3} \right )}{4x^{2}y^{-4}}$.

The sum of the first 6 terms of a sequence is 39.75. Also, the 7^th term is $latex \frac{6}{5}$ times the 2^nd term. Find the first term and the common difference of the sequence.

Let $latex log_{8}A=x, \, log_{8}B=y, \, log_{8}C=z$. Express $latex log_{8}\left ( \frac{A^{2}}{B^{3}C} \right )^{4}$ in terms of x, y and z.

Consider the expansion of $latex \left ( \frac{1}{3}x – 2 \right ) ^{7}$. Find the coefficient of $latex  x^{3}$.