Finding 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 you want to find the $$100^{th}$$​ term, it is not necessary to find any of the previous terms. You should be able to find the $$n^{th}$$ term formula for a linear or a quadratic sequence.

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

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Find the nth term formula for a linear sequence

A linear or arithmetic sequence is one where the difference 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 1

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

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.

$$\begin {array} {c|c c c c c c c}n(position) & 1 && 2 && 3 &&4 \\ \hline n^{th} \space term & 2 && 5 && 8 && 11 \\difference && \xrightarrow[+3]{} && \xrightarrow[+3]{} &&\xrightarrow[+3]{} \\3n & 3 && 6 && 9 && 12 \\3n-1 & 2 && 5 && 8 && 11\end {array}$$

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

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

Finding the nth term for a quadratic sequence - higher only

A quadratic sequence has its $$n^{th}$$ term formula in quadratic form as

​$$n^{th} \space term = an^2+bn+c$$​​

where $$a, b$$ and $$c$$ are constants to be found. You can find the $$n^{th}$$ term formula by putting the sequence and their positions $$n$$ into a table. A quadratic sequence can be identified by looking at adjacent terms in the sequence. If the difference between terms goes up in equal steps, then it is a quadratic sequence.

Example 2

​$$\begin{array}{c c c c c c c c c}5 & & 8 & & 13 & & 20 & & 29 \\& \xrightarrow[+3]{} && \xrightarrow[+5]{} && \xrightarrow[+7]{} && \xrightarrow[+9]{} \end{array}$$​​

​The adjacent terms go up in increasing steps, so this is a quadratic sequence.

PROCEDURE

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Example 3

Find the $$n^{th}$$ term formula for the sequence

​$$5, 11, 19, 29, 41$$​​

Start by filling the rows in a table with the first and second row of differences

​$$\begin{array}{c | c c c c c c c c c}n &1 && 2 && 3 && 4 &&5 \\ \hline n^{th} \space term & 5 && 11 && 19 && 29 && 41 \\d1 && +6 && +8 &&+10 && +12 \\d2 &&&+2 &&+2 && +2\end{array}$$​​

Form the equations from the table.

​$$\begin{array}{c | c c c c c c c c c}n &1 && 2 && 3 && 4 &&5 \\ \hline n^{th} \space term & \underbrace{5}_\text{a+b+c} && 11 && 19 && 29 && 41 \\d1 && \underbrace{+6}_\text{3a+b} && +8 &&+10 && +12 \\d2 &&&\underbrace{+2}_\text{2a} &&+2 && +2 \\\end{array}$$​​

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$$\begin {aligned}2a&=2 \\3a+b&=6 \\a+b+c&=5\end {aligned}$$​​

Solve the equations for $$a, b$$ and $$c$$.​

​$$\begin {aligned}2a&=2 \\a&=1 \\ \\3a+b&=6 \\3 \times 1 + b &=6 \\3+b &= 6\\b&= 3 \\ \\a+b+c&=5 \\1+ 3 + c &=5 \\4 +c &=5 \\c &=1\end {aligned}$$​​

Substitute the values of $$a, b$$ and $$c$$ into the formula $$n^{th} \space term = an^2+bn+c$$ to find the formula.​

​$$\underline {n^{th}\space term = n^2+3n+1}$$​​