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:
$u_{n}=3n-1$

Consider the illustrated pentagons and matches below:

A pentagon is made using matches.

By adding more matches, a row of two pentagons is formed.

Continuing to add matches, a row of three pentagons can be formed.

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:
$u_{n}=4n+1$

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} $

The general term of a number sequence is one of many ways of defining sequences.
Consider the tower of bricks. The first row has five bricks on top of the pile, the second row has six bricks, and the third row has seven bricks. If $ T_n $ represents the number of bricks in row $ n $ (from the top) then $ T_1 = 5, \ T_2 = 6, \ T_3 = 7, \cdots $
This sequence can be specified by using an explicit formula $ T_n = n + 4 $ is the general term or $n$th term formula for $ n = 1, 2, 3, 4, \cdots $.
A symbol with a subscript represents a sequence’s general term or nth term, for example, $ u_n, \ T_n, \ 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.

Question 1

A sequence is defined by $ u_n = 2n-3 $. Find $ u_9 $.

$ \begin{aligned} \displaystyle
u_9 &= 2 \times 9-3 \\
&= 15
\end{aligned} $

Question 2

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

$ u_{5} = 4 \times 5 + 2 = 22 $

Question 3

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

$ \begin{align} \displaystyle
T_{1} = 3 \times 1 + 7 = 10 \\
T_{2} = 3 \times 2 + 7 = 13 \\
T_{3} = 3 \times 3 + 7 = 16 \\
T_{4} = 3 \times 4 + 7 = 19 \\
T_{5} = 3 \times 5 + 7 = 22
\end{align} $

Question 4

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

$ \begin{align} \displaystyle
1^2 &= 1 \\
2^2 &= 4
\end{align} $

Question 5

A sequence is defined by $ T_n = n^2 + 1 $. Find the first four terms.

$ \begin{aligned} \displaystyle
T_1 &= 1^2 +1 = 2 \\
T_2 &= 2^2 +1 = 5 \\
T_3 &= 3^2 +1 = 10 \\
T_4 &= 4^2 +1 = 17
\end{aligned} $

Question 6

A sequence is defined by $ A_n = 2^n $. Find the first four terms.

$ \begin{aligned} \displaystyle
A_1 &= 2^1 = 2 \\
A_2 &= 2^2 = 4 \\
A_3 &= 2^3 = 8 \\
A_4 &= 2^4 = 16
\end{aligned} $

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