Arithmetic Sequences

In this lesson you will: Find the common difference and use it to write both a recursive rule and an explicit rule; Translate an arithmetic sequence between recursive and explicit form; Use an arithmetic sequence to answer a question about a situation.

The same step, every time

A sequence is arithmetic when you get each term by adding the same number to the term before it. That number is the common difference, written d.

For 7,11,15,19 the differences are 11-7=4, 15-11=4 and 19-15=4. Check all of them, not just the first pair. A sequence like 2,6,12,20 starts with a difference of 4 and then abandons it.

The explicit rule counts steps, not terms

To travel from a_1 to a_n you take n-1 steps. Four terms means three gaps. Ten terms means nine gaps. Count the spaces between your fingers if it helps.

Each step adds d, so the explicit rule adds d exactly n-1 times.

An arithmetic sequence is a line with most of it erased

Expand the explicit rule: a_n=a_1+(n-1)d=dn+(a_1-d). That is mx+b with the letters renamed. The common difference is the slope, and a_1-d is the value the line would have at n=0.

For 7,11,15,19,... this gives a_n=4n+3. Every term sits on the line y=4x+3, and the sequence is that line sampled at whole-number inputs and nowhere else.

Common mistake — Multiplying by n instead of by n minus 1

Asked for the tenth term of 7,11,15,19,..., a student writes 7+10(4)=47.

It feels right because the question says \"tenth\" and there is a 10 available to use. The number 47 is even a term of the sequence — it is the eleventh — so nothing about it looks alarming.

But getting from term 1 to term 10 takes nine steps, not ten. The correct computation is 7+9(4)=43.

The harsher version of the same slip drops a_1 altogether and writes 10(4)=40, which forgets where the sequence started.

How to catch it: Test your formula at n=1. It must return the first term. 7+(1-1)(4)=7 passes. 7+1(4)=11 fails, and it fails in about three seconds, before the wrong answer has cost you anything.