The first term of a geometric progression is $32$ and the fourth term is $\frac{1}{2}$. Find the sum to infinity of the progression.

The seventh term of an arithmetic progression is $19$ and the sum of the first twelve terms is $224$ . Find the fourth term.

A geometric progression has first term $3$ and common ratio $r$. A second geometric progression has first term $2$ and common ratio $\frac{1}{5}r$. The two progressions have the same sum to infinity $S$. Find the value of $r$ and the value of $S$.

A geometric progression has first term $a$, common ratio $r$ and sum to infinity $S$. A second geometric progression has first term $5a$ , common ratio $3r$ and sum to infinity $10S$. Find the value of $r$.[Write your answer as decimal]

An arithmetic progression has first term $-4$. The $n^{\mathrm{th}}$ term is $8$ and the $2n^{\mathrm{th}}$ term is $20.8$. Find the value of $n$.

A television quiz show takes place every day. On day $1$ the prize money is $\$1000$. If this is not won the prize money is increased for day $2$ .The prize money is increased in a similar way every day until it is won. The television company considered the following model for increasing the prize money.
Model : Increase the prize money by $\$1000$ each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is $5\%$ of the value of the prize on that day. After $40$ days the prize money has still not been won. Calculate the total amount donated to charity.[Write your answer as numerical value only]

A television quiz show takes place every day. On day $1$ the prize money is $\$1000$. If this is not won the prize money is increased for day $2$ .The prize money is increased in a similar way every day until it is won. The television company considered the following model for increasing the prize money.
Model : Increase the prize money by $10\%$ each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is $5\%$ of the value of the prize on that day. After $40$ days the prize money has still not been won. Calculate the total amount donated to charity.[Write your answer as numerical value only after correcting up to two decimals]

The first two terms of an arithmetic progression are $1 \text{and} {\cos }^{2}x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a-b{\sin }^{2}x$, where $a$ and $b$ are constants to be found.

The first two terms of a geometric progression are $1$ and $\frac{1}{3}{\tan }^2\theta$ respectively. where $0<\theta <\frac{1}{2}\pi$. Find the set of values of $\theta$ for which the progression is convergent.