# Arithmetic Sequence Problems

An arithmetic sequence is a sequence where there is a common difference between any two successive terms.

$$\require{color} \color{red} u_{n} = u_{1}+(n-1)d$$

where $\require{color} \color{red} u_{1}$ is the first term and $\require{color} \color{red}d$ is the common difference of the arithmetic sequence.

### Example 1

A city is studies and found to have a population of $5000$ in the first year of the study. The population increases by $200$ each year after that.

(a)   Write down a rule for the population in month $n$ of the study.

(b)   When will the population double in size?

### Example 2

For the arithmetic sequence $\{21,x,y,36\}$, find the values of $x$ and $y$.

### Example 3

Find the value of $x$ such that $\{\cdots,x,3x+4,10x-7,\cdots\}$ forms an arithmetic sequence.