The arithmetic sequence

In an arithmetic sequence each term is greater than the term before it by the same number, $\displaystyle d$ – the common difference:

• $\displaystyle 5, 8, 11, 14, \ldots$ is an arithmetic sequence with $\displaystyle {a_1}={5}$ and $\displaystyle {d}={3}$.
• $\displaystyle 20, 16, 12, \ldots$ is an arithmetic sequence with $\displaystyle {d}={-4}$: it decreases.
• $\displaystyle 7, 7, 7, \ldots$ is an arithmetic sequence with $\displaystyle {d}={0}$: a constant sequence.

From the first term to the term in place $\displaystyle n$ there are $\displaystyle n-1$ steps, and each step adds $\displaystyle d$:

In the sequence $\displaystyle 5, 8, 11, \ldots$:

• $\displaystyle {a_{30}}={5+29\cdot3}={92}$.
• The formula by place: $\displaystyle {a_n}={5+(n-1)\cdot3}={3n+2}$.

The formula $\displaystyle {a_n}={3n+2}$ is a linear function of $\displaystyle n$. So the points $\displaystyle (n, a_n)$ lie on the line $\displaystyle {y}={3x+2}$ (dashed), and its slope is the difference $\displaystyle {d}={3}$:

In an arithmetic sequence each term (but the first) is the arithmetic mean of its two neighbours – hence the name “arithmetic sequence”:

• The neighbour after it is greater by $\displaystyle d$, and the neighbour before it smaller by $\displaystyle d$, so their mean is the term itself.
• Three numbers $\displaystyle x, y, z$ (in this order) are neighbouring terms of an arithmetic sequence exactly when $\displaystyle {2y}={x+z}$.

In the sequence $\displaystyle 12, 19, 26, \ldots$ ($\displaystyle {a_1}={12}$, $\displaystyle {d}={7}$):

• Is 404 a term? $\displaystyle {12+(n-1)\cdot7}={404}$, that is $\displaystyle {(n-1)\cdot7}={392}$, $\displaystyle {n-1}={56}$ and $\displaystyle {n}={57}$: 404 is the 57th term.
• Is 100 a term? $\displaystyle {12+(n-1)\cdot7}={100}$ gives $\displaystyle {n-1}={\frac{88}{7}}$, which is not a whole number: 100 is not a term.
• The finite sequence $\displaystyle 12, 19, 26, \ldots, 404$ has 57 terms: the number of terms is the place of the last term.

• To prove that a sequence is arithmetic, show that $\displaystyle a_{n+1}-a_n$ is the same number for every $\displaystyle n$. In the sequence $\displaystyle {a_n}={7-2n}$: $\displaystyle {a_{n+1}-a_n}={(7-2(n+1))-(7-2n)}={-2}$, so the sequence is arithmetic and $\displaystyle {d}={-2}$.
• To show that a sequence is not arithmetic one counterexample is enough: two differences that are not equal. In the sequence $\displaystyle {a_n}={n^2}$: $\displaystyle {a_2-a_1}={3}$ but $\displaystyle {a_3-a_2}={5}$.
• Checking a few terms, even a thousand terms on a calculator, is not a proof that a sequence is arithmetic.

The arithmetic sequence: $\displaystyle 7, 11, 15, \ldots$
Calculate $\displaystyle a_{40}$.

The sequence: $\displaystyle {a_n}={5-3n}$.
Prove that the sequence is arithmetic, and find its difference.

How many terms has the arithmetic sequence $\displaystyle 3, 10, 17, \ldots, 248$?

The numbers $\displaystyle x+1$, $\displaystyle 3x$, $\displaystyle 4x+2$ (in this order) are three neighbouring terms of an arithmetic sequence.
Find $\displaystyle x$ and the three terms.

The sequence: $\displaystyle {a_n}={n^2+1}$.
Is the sequence arithmetic? Explain.