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The nth term

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Explainer Video

Tutor: Meera

Summary

The nth term

In a nutshell

Sequences defined using an $n^{th}$ term formula help us find the sequence from the position of the terms in the sequence. The $n^{th}$ term method is more useful, as if we want to find the 100th term, it is not necessary to find any of the previous terms.

Generating a sequence

The $n^{th}$ term formula can be used to generate a sequence. Start by substituting $n=1$ into the formula to find the first term. Then substitute $n=2$ for the second term, $n=3$ for the third term and so on.

Examples

NTH TERM FORMULA	SEQUENCE
$n^{th} term = 4n+4$	$8,12, 16, 20,24$
$n^{th} term = 8-6n$	$2, -4, -10, -16, -22$
$n^{th} term = n^2+2$	$3, 6, 11, 18, 27$

Find the nth term formula for an arithmetic sequence

An arithmetic sequence is one where the differences between adjacent terms are always the same. The term-to-term rule would be to $+$ or $-$ the same number each time.

It is possible to find the $n^{th}$ term formula for an arithmetic sequence.

Example

Find the $n^{th}$ term formula for the sequence $2, 5, 8, 11$ .

Answer

Start by putting the sequence, and their positions $n$ into a table. Work out the difference each time, in this case the difference is $3$ , so add another row in the table for $3n$ . Then think about how to get from $3n$ to the sequence, in this case each term in our sequence is $1$ less than the $3n$ row, so the formula is $\underline{n^{th} \space term = 3n-1}$ .

$n$ (the position of each term)	$1$	$2$	$3$	$4$
$n^{th} \space term$ (the sequence)	$2$	$5$	$8$	$11$
the difference is $3$ so use $3n$	$3$	$6$	$9$	$12$
each term is $1$ less than $3n$ , so use $3n-1$	$2$	$5$	$8$	$11$

$\underline {n^{th} \space term = 3n-1}$

Note: Always check the formula, by substituting values for $n$ to see if it works.

Learn with Basics

Length:

Unit 1

Linear number sequences

Unit 2

Number patterns and sequences

Jump Ahead

Unit 3

The nth term

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you find the nth term formula for an arithmetic sequence?

Start by finding the difference between term, this is the number that goes with n. For example, if the difference is +3, then use 3n. Then either add or subtract a constant until it matches the sequence.

Is nth term better than term-to-term rule?

The nth term method of defining a sequence can be more useful than the term-to-term rule, as it does not rely on finding any other previous terms in the sequence.

What is an nth term formula?

An nth term formula is a way of defining a sequence. The 'n' stands for the position of the term in the sequence. We can substitute different values of n to find the term.

Home

Maths

Formulae and equations

The nth term

Videos

Summary

Exercises

Select Lesson

Statistics

Qualitative and quantitative data

Representing data: Graphs and charts

Mean, median, mode and range

Frequency tables

Averages from frequency tables

Averages from grouped frequency tables

Scatter graphs

Probability

Calculating probability

Equal and unequal probabilities

Relative frequency

Listing outcomes

Venn diagrams

Trigonometry

Pythagoras’ theorem

Trigonometric ratios: SOH CAH TOA

Finding missing lengths in a triangle

Finding missing angles in a triangle

Drawing shapes

Transformations

Constructions

Constructing triangles

Lines and angles

Types of angles

Angle rules

Interior and exterior angles

Parallel lines

Measures

Area and perimeter: Triangles and quadrilaterals

Area of compound shapes

Volume of prisms

Properties of shapes

Symmetrical shapes and rotational symmetry

Congruent shapes

Similar shapes

Nets and surface area

Shapes

Quadrilaterals: Types and properties

Triangles and regular polygons: Types and properties

Circles: Area and circumference

3D shapes: Identification and properties

Rates of change

Percentage change

Conversion factors

Imperial units

Solving best buy problems

Reading timetables

Maps and scale drawings

Speed: Formula and units

Density: Formula and units

Proportion

Ratio

Other graphs

Interpreting graphs

Solving simultaneous equations using graphs

Quadratic graphs

Travel graphs: Distance-time graphs

Conversion graphs

Straight line graphs

X and Y coordinates

Straight line graphs

Drawing straight line graphs

Finding the gradient

Equation of a straight line: y = mx + c

Drawing a graph of a linear equation

Estimating values using linear graphs

Formulae and equations

Substituting numbers into an expression

Writing and using formulae

Solving equations

Rearranging formulae

Number patterns and sequences

The nth term

Inequalities: Greater than or less than

Simultaneous equations

Algebraic manipulation

Algebraic notation

Algebraic terms

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Expanding double brackets: FOIL and multiplication grid

Fractions, decimals and percentages

Converting between fractions, decimals and percentages

Fractions

Fraction operations: Add, subtract, multiply, divide

Percentages

Rounding: Decimal places and significant figures

Rounding errors and estimating

Number calculations

Ordering numbers: Whole numbers and decimals

Addition and subtraction using the column method

Multiplying and dividing by powers of 10

Adding and subtracting decimals

Methods for multiplication and division

Order of operations: BIDMAS

Types of numbers

Understanding place value

Negative numbers: Add, subtract, multiply, divide

Special types of numbers

Multiples, factors and prime factors

Lowest common multiple and highest common factor

Square roots and cube roots

Writing indices and index laws

Standard form

Explainer Video

Tutor: Meera

Summary

The nth term

In a nutshell

Generating a sequence

Examples

NTH TERM FORMULA	SEQUENCE
$n^{th} term = 4n+4$	$8,12, 16, 20,24$
$n^{th} term = 8-6n$	$2, -4, -10, -16, -22$
$n^{th} term = n^2+2$	$3, 6, 11, 18, 27$

Find the nth term formula for an arithmetic sequence

An arithmetic sequence is one where the differences between adjacent terms are always the same. The term-to-term rule would be to $+$ or $-$ the same number each time.

It is possible to find the $n^{th}$ term formula for an arithmetic sequence.

Example

Find the $n^{th}$ term formula for the sequence $2, 5, 8, 11$ .

Answer

$n$ (the position of each term)	$1$	$2$	$3$	$4$
$n^{th} \space term$ (the sequence)	$2$	$5$	$8$	$11$
the difference is $3$ so use $3n$	$3$	$6$	$9$	$12$
each term is $1$ less than $3n$ , so use $3n-1$	$2$	$5$	$8$	$11$

$\underline {n^{th} \space term = 3n-1}$

Note: Always check the formula, by substituting values for $n$ to see if it works.

Learn with Basics

Length:

Unit 1

Linear number sequences

Unit 2

Number patterns and sequences

Jump Ahead

Unit 3

The nth term

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you find the nth term formula for an arithmetic sequence?

Is nth term better than term-to-term rule?

The nth term method of defining a sequence can be more useful than the term-to-term rule, as it does not rely on finding any other previous terms in the sequence.

What is an nth term formula?

An nth term formula is a way of defining a sequence. The 'n' stands for the position of the term in the sequence. We can substitute different values of n to find the term.