Patterns and the nth Term: Quick Revision

A brief note on Patterns and the nth Term. Read it before a test. Try each question at the end before you open its answer.

Arithmetic progressions

• An AP adds the same number $d$ to each term: $a, a + d, a + 2d, \dots$
• Common difference: $d =$ a term minus the term just before it. It can be positive, negative or zero.
• To test a list, every difference between neighbours must be equal.
• A finite AP has a last term $l$; an infinite AP goes on for ever.

The nth term

• $a_n = a + (n - 1)d$.
• Knowing three of $a$, $d$, $n$ and $a_n$ gives the fourth.
• A number is a term only if $n$ comes out a whole number of at least $1$.
• The $n$th term from the end is $l - (n - 1)d$.

Problems

• Two terms given: subtract them. The number of steps is the difference of the term numbers.
• Three numbers in an AP: $a - d$, $a$, $a + d$. Their sum is $3a$.
• Counting numbers of a kind: find the first and the last, then solve for $n$.

A worked example

The $4$th term of an AP is $17$ and the $9$th term is $42$. Find its $30$th term.

1. Steps. From the $4$th to the $9$th term is $5$ steps: $5d = 25$, so $d = 5$.
2. First term. $a = 17 - 3 \times 5 = 2$.
3. The 30th term. $a_{30} = 2 + 29 \times 5 = 147$.

Practice with new numbers

A program to try

The program finds a term of the AP $6, 11, 16, \dots$ and searches for the term number of a value.

Mistakes to avoid

• Subtracting the wrong way: $d$ is a term minus the term before it.
• Adding $d$ $n$ times: the $n$th term adds it only $n - 1$ times.
• Forgetting the last $+1$ when you solve for $n$.
• Accepting a decimal $n$: then the number is not a term.

Check yourself

1. Is $8, 5, 2, -1$ an AP? If so, give $d$.

2. Find the $16$th term of $3, 7, 11, \dots$

3. Is $100$ a term of $1, 4, 7, \dots$?

4. Three numbers in an AP add up to $45$. What is the middle one?