A sequence and a formula by place

A sequence is a list of numbers in a certain order: it has a first term, a second term, a third term and so on.

• Each number in the list is a term of the sequence.
• The place of a term is a natural number: $\displaystyle 1, 2, 3, \ldots$
• $\displaystyle a_n$ is the term in place $\displaystyle n$: in the sequence of the odd numbers $\displaystyle 1, 3, 5, 7, \ldots$ the first term is $\displaystyle {a_1}={1}$, and the fourth term is $\displaystyle {a_4}={7}$.
• The terms themselves need not be natural numbers: in the sequence $\displaystyle \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$ every term is a fraction.

A formula by place gives a term from its place. In the sequence of the odd numbers $\displaystyle {a_n}={2n-1}$:

• A sequence is a function: each natural number $\displaystyle n$ has exactly one value, $\displaystyle a_n$. So $\displaystyle a(n)$ is sometimes written for $\displaystyle a_n$.
• The difference from the functions you know: the variable $\displaystyle n$ takes natural values only.
• So the graph of a sequence is separate points $\displaystyle (n, a_n)$, not a line:

In stage $\displaystyle n$ there is a square of side $\displaystyle n+1$ squares, and the squares along its edge are shaded. $\displaystyle a_n$ is the number of shaded squares: $\displaystyle 4, 8, 12, 16, \ldots$

The formula can be found in two ways:

• Four sides of $\displaystyle n+1$ squares each, and each corner square was counted twice: $\displaystyle {a_n}={4(n+1)-4}={4n}$.
• The edge splits into four strips of $\displaystyle n$ squares (each strip starting at a different corner): $\displaystyle {a_n}={4n}$.

Both ways give the same formula, and the table confirms it: $\displaystyle {a_3}={12}$.

• A sequence is increasing if each term is greater than the term before it: $\displaystyle a_{n+1}>a_n$ for every $\displaystyle n$.
• A sequence is decreasing if each term is smaller than the term before it: $\displaystyle a_{n+1}<a_n$ for every $\displaystyle n$.
• To decide, work out the difference $\displaystyle a_{n+1}-a_n$ and look at its sign.

An example: $\displaystyle {a_n}={n^2-4n}$, that is $\displaystyle -3, -4, -3, 0, 5, \ldots$

• For $\displaystyle {n}={1}$ the difference is negative: $\displaystyle a_2<a_1$.
• For $\displaystyle n\geq2$ the difference is positive: from the second term on, the sequence increases.
• The sequence as a whole is neither increasing nor decreasing.

In the sequence $\displaystyle {a_n}={(-1)^n\cdot n}$ the terms are $\displaystyle -1, 2, -3, 4, -5, \ldots$

• $\displaystyle (-1)^n$ equals $\displaystyle 1$ when $\displaystyle n$ is even, and $\displaystyle -1$ when $\displaystyle n$ is odd.
• So the terms are positive and negative in turn: the sequence has alternating signs.
• Such a sequence is neither increasing nor decreasing.

In which place of the sequence $\displaystyle {a_n}={n^2-4n}$ is the term 45? We solve an equation in $\displaystyle n$:

• $\displaystyle {n^2-4n-45}={0}$, that is $\displaystyle {(n-9)(n+5)}={0}$, so $\displaystyle {n}={9}$ or $\displaystyle {n}={-5}$.
• $\displaystyle {n}={-5}$ is not the place of a term: a place is a natural number.
• 45 is the ninth term: $\displaystyle {a_9}={81-36}={45}$.

A number whose equation has only solutions that are not natural is not a term of the sequence.

The sequence: $\displaystyle {a_n}={3n+2}$.

Calculate $\displaystyle a_{10}$.

Is 100 a term of the sequence? Solve an equation, and explain.

The sequence: $\displaystyle {a_n}={\frac{n}{n+1}}$.

Write the first four terms.

Simplify $\displaystyle a_{n+1}-a_n$, and decide whether the sequence increases or decreases.

The sequence: $\displaystyle {a_n}={n^2-6n+5}$.
Which terms of the sequence are negative? Solve an inequality, and write their places.

In the picture of the squares above, $\displaystyle b_n$ is the number of squares not shaded in stage $\displaystyle n$.

Write a formula by place for $\displaystyle b_n$, and explain it.

In which stage are there 400 squares not shaded?