General Term of a Number Sequence

Sequences may be defined in one of the following ways:
  • listing all terms of a finite sequence:
    $2, 5, 8, 11, 14, 17$
  • listing the first few terms and assuming that the pattern represented continuous indefinitely:
    $2, 5, 8, \cdots $
  • giving a description in words:
    $\textit{Starts at 2, and each term is 3 more than the previous term}$
  • using a formula which represents the general term of $n$th term:
Consider the illustrated pentagons and matches below.

A pentagon is made using matches. Pentagon 1 and 5 matches in General Term of a Number Sequence
By adding more matches, a row of two pentagons is formed. Pentagon 2 and 9 matches in General Term of a Number Sequence
Continuing to add matches, a row of three pentagons can be formed. Pentagon 3 and 13 matches in General Term of a Number Sequence
If $u_{n}$ represents the number of matches in $n$ pentagons, then $u_{1}=5,u_{2}=9,u_{3}=13$ and so on.

This sequence can be specified by:
  • listing terms:
    $5, 9, 13, \cdots$
  • using words:
    $\textit{The first term is 5 and each term is 4 more than the previous term.}$
  • using an explicit formula:

General Term

The $\textit{General Term}$ or $n$th term of a sequence is represented by a symbol with a subscript.
For example, $u_{n}$, $T_{n}$ or $A_{n}$. The general term is defined for $n=1,2,3,4, \cdots$.

$\{u_{n}\}$ represents the sequence that can be generated by using $u_{n}$ as the $n$th term.
For example, $\{3n-1\}$ generates the sequence:
\( \begin{align} \displaystyle u_{1} &= 3 \times 1 - 1 = 2 \\ u_{2} &= 3 \times 2 - 1 = 5 \\ u_{3} &= 3 \times 3 - 1 = 8 \\ \end{align} \)

Example 1

A sequence is defined by $u_{n}=4n+2$. Find $u_{5}$.

Example 2

Find the first five terms of $T_{n}=3n+7$.

Example 3

Find the first two terms of $\{n^2\}$.

Example 4

Find the first three terms of $u_{n}=2^n$.

Example 5

Find $T_{4}$ of $T_{n+1}=n^2-1$.