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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

The General Term of an Arithmetic Sequence

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 29 s
  • British Columbia
  • BC10-MA-21

Using the sequence 7, 11, 15, …, the video names terms, term numbers and the common difference d = 4, then shows that reaching each term takes one fewer jump of 4 than its term number, which leads to tₙ = t₁ + (n − 1)d and t₂₀ = 7 + 19 × 4 = 83 (and warns that 7 + 20 × 4 = 87 is the 21st term). It then simplifies the rule to tₙ = 4n + 3, checks it with n = 1 and n = 20, and works backwards to show that 103 is the 25th term, checked as 7 + 24 × 4 = 103.

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In one sentence

The question this lesson answers

Here is a sequence: 7, 11, 15, and so on. What is its 20th term?

Each number in the list is a term: t sub 1, t sub 2, t sub 3, and so on.

Transcript

What the lesson says

Here is a sequence: 7, 11, 15, and so on. What is its 20th term?

Each number in the list is a term: t sub 1, t sub 2, t sub 3, and so on.

Here n is the term number, so t sub n is the nth term.

In an arithmetic sequence, we add the same amount each time.

That amount is the common difference, d. 11 minus 7 is 4, and 15 minus 11 is 4, so d is 4.

Start at t sub 1, which is 7. To reach t sub 2, add one 4. To reach t sub 3, add two 4s.

The number of 4s is one less than the term number, so t sub 20 needs 19 of them.

This gives the general term: t sub n equals t sub 1, plus n minus 1, times d.

For the 20th term, put in n equals 20, t sub 1 equals 7, and d equals 4.

20 minus 1 is 19, 19 times 4 is 76, and 7 plus 76 is 83. The 20th term is 83.

Careful: multiply by n minus 1, not by n. 7 plus 20 times 4 is 87, the 21st term.

We can simplify the general term. Put in t sub 1 equals 7 and d equals 4.

Expand the brackets to get 7 plus 4n minus 4, then collect like terms: 4n plus 3.

Check: 4 times 1 plus 3 is 7, and 4 times 20 plus 3 is 83.

Now work backwards: which term is 103?

Set 4n plus 3 equal to 103. Subtract 3 to get 4n equals 100, then divide by 4: n equals 25.

Check: t sub 25 is 7 plus 24 times 4, which is 7 plus 96, or 103.

To sum up: the nth term is the first term plus n minus 1 common differences.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Linear relations: arithmetic sequencesBC10-MA-21

Previous: Arithmetic sequences: first term and common differenceNext: Arithmetic Sequences Are Linear: The Common Difference Is the Slope

The outcomes and strands follow the British Columbia curriculum.

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