# Arithmetic Sequence Problems

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

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

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

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

\( \begin{align} \displaystyle

u_{n} &= 5000+(n-1)\times 200 \\

&= 5000 + 200n-200 \\

u_{n} &= 200n + 4800 \\

\end{align} \)

(b) β When will the population double in size?

\( \begin{align} \displaystyle

200n + 4800 &= 10000 \\

200n &= 5200 \\

n &= 5200 \div 200 \\

&= 26 \\

\end{align} \)

Thus the population will double in the $26$^{th} month.

## Example 2

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

\( \begin{align} \displaystyle

u_{1} &= 21 \\

u_{2} &= 21 + d = x \cdots (1) \\

u_{3} &= 21 + 2d = y \cdots (2) \\

u_{4} &= 21 + 3d = 36 \cdots (3) \\

3d &= 36-21 \cdots (3) \\

3d &= 15 \\

d &= 5 \\

x &= 21 + 5 &\text{substitute } d = 5 \text{ into } (1) \\

&= 26 \\

y &= 21 + 2 \times 5 &\text{substitute } d = 5 \text{ into } (2) \\

&= 21+10 \\

&= 31 \\

\therefore x &= 26 \text{ and } y=31 \\

\end{align} \)

## Example 3

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

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

\( \begin{align} \displaystyle

(3x+4)-(x) &= (10x-7)-(3x+4) \\

3x+4-x &= 10x-7 – 3x-4 \\

2x+4 &= 7x-11 \\

2x-7x &= -11-4 \\

-5x &= -15 \\

\therefore x &= 3

\end{align} \)

**β **Unlock your full learning potentialβdownload our expertly crafted slide files for free and transform your self-study sessions!

**β **Discover more enlightening videos by visiting our YouTube channel!

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

## Responses