Find the sum of the first eight terms of the following geometric series,

$1-2+4-8+\cdots$

Find the sum of the first eight terms of the following geometric series,

$243+162+108+72+\cdots$

The first four terms of a geometric progression are $0.5,1,2\text{ and }4$. Find the smallest number of terms that will give a sum greater than $1000000$.

A ball is thrown vertically upwards from the ground. The ball rises to a height of $8$ m and then falls and bounces. After each bounce it rises to $\frac{3}{4}$ of the height of the previous bounce. Write down an expression for the height that the ball rises after the $n^{\mathrm{th}}$ impact with the ground.

A ball is thrown vertically upwards from the ground. The ball rises to a height of $8$ m and then falls and bounces. After each bounce it rises to $\frac{3}{4}$ of the height of the previous bounce. Find the total distance that the ball travels from the first throw to the fifth impact with the ground.

The second term of a geometric progression is $24$ and the third term is $12(x+1)$. Find, in terms of $x$, the first term of the progression.

The second term of a geometric progression is $24$ and the third term is $12(x+1)$. Given that the sum of the first three terms is $76$ , find the possible values of $x$.

The third term of a geometric progression is nine times the first term. The sum of the first four terms is $k$ times the first term. Find the possible values of $k$.

A company makes a donation to charity each year. The value of the donation increases exponentially by $10\%$ each year. The value of the donation in $2010$ was $\$10000$. Find the value of the donation in $2016$ .