The sum of the second and third terms in a geometric progression is $30$ . The second term is $9$ less than the first term. Given that all the terms in the progression are positive, find the first term.

Three consecutive terms of a geometric progression are $x,4\text{ and }x+6$. Find the possible values of $x$.

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

$3+6+12+24+\cdots$

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

$128+64+32+16+\cdots$

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.