Arithmetic Sequence Formulae and Subscript Notation

From positions to terms

A sequence is an ordered list, so the position of a term matters. We usually number the positions $1,2,3,\ldots$ and use $n$ for a general position. If the sequence is called $u$, then

• $u_1$ means the first term;
• $u_2$ means the second term;
• $u_n$ means the term in position $n$.

The subscript is a label for the position. Thus $u_3$ is the third term, not $u\times3$, and $u_{n+1}$ is the term immediately after .

A position-to-term rule calculates any term directly from its position. For example,

$u_n=5n-2$

gives $u_1=3$, $u_2=8$ and $u_3=13$. It can also find a distant term without generating all the earlier ones: substitute its position for .

What makes a sequence arithmetic?

An arithmetic sequence changes by the same amount at every step. This fixed amount is the common difference, usually denoted by $d$.

For $6,10,14,18,\ldots$, the common difference is $4$. For $20,17,14,11,\ldots$, it is $$ because is added each time. A negative common difference therefore describes a decreasing sequence.

If the first term is $a$, the terms can be written as

$a,\quad a+d,\quad a+2d,\quad a+3d,\ldots$

The first term has had no differences added, the second has had one, and the third has had two. By position $n$, exactly $n-1$ copies of the common difference have been added. This gives the general formula

$u_n=a+(n-1)d.$

Expanding it gives an equivalent form:

$u_n=dn+(a-d).$

This explains why the coefficient of $n$ is the common difference.

Deriving an nth-term formula

Consider

$3,7,11,15,19,\ldots$

The common difference is $4$, so begin with $4n$. The sequence generated by $4n$ is

$4,8,12,16,20,\ldots$

Each of these is one greater than the required term, so subtract $1$:

$u_n=4n-1.$

This is the same result obtained from $a+(n-1)d$:

$u_n=3+(n-1)\times4=4n-1.$

Always check the rule. Substituting $n=1$ gives $4(1)-1=3$, and substituting $n=2$ gives $4(2$.

The method also works for a decreasing sequence. For

$40,37,34,31,\ldots$

the common difference is $-3$. Starting with $-3n$ produces $-3,-6,-9,-12,\ldots$. Each required term is $43$ greater, so

$u_n=43-3n.$

Equivalently,

$u_n=40+(n-1)(-3)=43-3n.$

Position-to-term notation

A position-to-term rule may be written using a subscript, such as

$x_n=n+2.$

Here the input is the position $n$, and the output is the term $x_n$. For example, $x_5=5+2=7$. This rule determines every term directly and does not require knowledge of the preceding term.

The letter naming a sequence can vary: $x_n$, $u_n$ and $a_n$ can all mean “the term in position ”. What matters is using the same letter consistently.

Term-to-term notation

Subscripts can also express a term-to-term rule. For example,

$x_{n+1}=2x_n-3$

means: to obtain the next term, double the current term and subtract $3$. If $x_1=4$, then

$x_2=2(4)-3=5,$

$x_3=2(5)-3=7,$

so the sequence begins $4,5,7,11,\ldots$.

A term-to-term rule normally needs an initial term such as $x_1=4$. Without it, $x_{n+1}=2x_n-3$ describes the relationship between consecutive terms but does not identify one unique sequence.

For an arithmetic sequence with common difference $d$, the term-to-term form is

$u_{n+1}=u_n+d,$

along with the first term. For example, the position-to-term rule $u_n=5n+2$ produces $7,12,17,22,\ldots$, so the same sequence can be described by

$u_1=7,\qquad u_{n+1}=u_n+5.$

The two forms serve different purposes: $u_n=5n+2$ jumps directly to any position, whereas $u_{n+1}=u_n+5$ builds the sequence one term at a time.