Mathematics, four units · Grade 11

How do you find the first term and the common difference from two given terms?

Build two equations. If the fifth and ninth terms are given, write each as the first term plus the number of differences before it, and subtract one equation from the other. The common difference falls out at once, because the gap between those terms contains exactly four of them, and substituting back gives the first term.

Learning objectives

Two numbers define everything

An arithmetic sequence adds the same amount at every step. That amount is the common difference, written d, and the sequence is fixed entirely by two numbers: the first term and d.

The general term is the first term plus d times the number of steps taken to reach it. Note that reaching the fifth term takes four steps and not five — this is the single commonest source of error in the topic.

Testing whether a sequence is arithmetic is easy: subtract each term from the one before it. If every difference is the same, it is arithmetic. If the ratios are the same instead, it is geometric, which is the next lesson.

Finding d from two terms

When two non-adjacent terms are given, the gap between them contains a known number of ds. Between the fifth and the ninth term there are four steps, so the difference in their values is four times d.

From there d comes out in a single division, with no system of equations and no substitution. Then go back to either given term and extract the first term.

The same thinking answers the reverse question: which position a given value occupies. Write the general term, set it equal to the value and solve for n. If n comes out fractional, the value is simply not in the sequence — and that is a complete answer.

The sum of the first terms

The sum of the first n terms is the number of terms times the average of the first and last. That is the form worth remembering, because it is also the easiest to explain to yourself.

The idea behind it: pair the first term with the last, the second with the second-to-last, and so on. Every pair has exactly the same total, and the number of pairs is half the number of terms.

If the last term is not given, compute it from the general term first. Using the sum formula with a guessed last term is the quickest way to lose an entire question.

Worked examples

  1. In an arithmetic sequence the fifth term is 17 and the ninth is 33. Find d and the first term.

    1. Between positions five and nine there are four steps
    2. 33 minus 17 is 16, and that is four times d
    3. So d is 4
    4. The fifth term is the first plus four ds: 17 equals a plus 16

    Answer: d is 4 and the first term is 1

  2. In that same sequence, what is the sum of the first 20 terms?

    1. The twentieth term: 1 plus 19 times 4, which is 77
    2. Average of first and last: 1 plus 77 over 2, which is 39
    3. Multiply by the number of terms

    Answer: 20 times 39, which is 780

  3. A hall has 12 seats in the first row and 3 more in each row after it. How many seats in 15 rows?

    1. This is arithmetic with first term 12 and d equal to 3
    2. The fifteenth row: 12 plus 14 times 3, which is 54
    3. Average: 12 plus 54 over 2, which is 33

    Answer: 15 times 33, which is 495 seats

Common mistakes

Multiplying by n instead of n minus one in the general term
Reaching the nth term takes n minus one steps. Multiplying by n shifts the whole sequence along by one place and produces an answer that looks entirely plausible.
Using the sum formula with a last term that was never computed
The formula needs the actual last term. If it is not given, find it from the general term first; guessing it carries the error straight into the final answer.
Concluding a sequence is arithmetic because it increases
Increasing is not enough. A geometric sequence increases, and so does a sequence of squares. Arithmetic means all the differences are equal, and that is checked rather than assumed.

What to remember

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