2.11.2 nth term
To find the \( n^{th} \) term \( (a_n) \) of a linear sequence
Find the common difference \( d \) between the terms
Put the first term and the common difference \( d \) in the formula
\( a_n=a_1+(n-1)d \)
To find the \( n^{th} \) term of a quadratic sequence
Make sure you are dealing with quadratic sequence
Work out the first and second order differences
Divide the second difference by \( 2 \) and multiply the result by \( n^2 \) ,\( an^2 \)
Write down some terms of \( an^2 \) with the sequence underneath
Work out the difference between them
Find the \( n^{th} \) term of the difference which is a linear sequence
Add them together to get the \( n^{th} \) term of the quadratic sequence
Worked example:
For the sequence \( 5, 7, 11, 17, 25, … \)
Find a formula for the \( n^{th} \) term.
Identify the sequence by finding difference between the terms
Sequence: \( 5, 7, 11, 17, 25, … \)
First difference: \( 2, 4, 6, 8, … \)
Second difference: \( 2, 2, 2, … \)
As second difference is constant, so the sequence is a quadratic sequence
Divide the second difference by \( 2 \) and multiply the result by \( n^2 \)
\( a=2\div 2=1 \)
\( an^2=1\times n^2=n^2 \)
Write down the \( n^2 \) and the sequence underneath
On the next line, write down the difference between \( n^2 \) and the sequence
\( n^2:1, 4, 9, 16, 25, … \)
Sequence:\( 5, 7,11,17,25, … \)
Difference:\( 4, 3, 2, 1, 0,… \)
Find the \( n^{th} \) term of the sequence of this difference by using \( a_n=a_1+(n-1)d \)
\( a_1=4 \), \( d=-1 \)
\( a_1 +(n-1)d=4+(n-1)\times -1 \)
\( =4+(-n+1) \)
\( =5-n \)
Now add \( an^2 \) and the \( n^{th} \) term of the difference to get the \( n^{th} \) term of the sequence
\( n^2-n+5 \)
Test yourself:
Question 1:
The nthterm of another sequence is \( 4n^2+n+3 \).
Find the \( 2^{nd} term \).
[1]
Find the value of \( n \) when the \( n^{th} \) term is \( 498 \).
[3]
Question 2:
Complete the table for the \( 5^{th} \) term and the \( n^{th} \) term of each sequence.
[11]
Question 3:
\( 0, 1, 1, 2, 3, 5, 8, 13, 21,… \)
This sequence is a Fibonacci sequence.
After the first two terms, the rule to find the next term is “add the two previous terms”.
Use this rule to complete each of the following fibonacci sequence.
[3]
Question 4:
These are first four diagrams of a sequence.
The diagrams are made from white dots and black dots
Complete the table for diagram \( 5 \) and diagram \( 6 \).
[2]
Write an expression, in terms of \( n \), for the number white dots in diagram \( n \).
[1]
The expression for the total number of dots in diagram \( n \) is \( \frac{1}{2}(3n^2-n) \).
Find the total number of dots in diagram \( 8 \).
[1]
Find an expression for the number of black dots in Diagram \( n \).
Give your answer in its simplest form.
[2]
\( T \) is the total number of dots used to make all of the first \( n \) diagrams.
\( T=an^3+bn^2 \)
Find the value of \( a \) and the value of \( b \) .
You must show all your working.
[5]
Question 5:
The table shows the first four terms in sequences \( A \), \( B \), and \( C \).
Complete the table.
[9]