EN
ScientistGrades 9–10⏱ 15 min

Patterns and sequences

Build arithmetic and geometric sequences as bars, find the rule for any term, and see doubling overtake adding.

Standards: F-IF.A.3F-BF.A.2

1Wonder

Option A: get 10today,then10 today, then 10 more than the day before, every day. Option B: get 1 cent today, then double it every day. After 30 days, which would you rather have?

2Explore

Pattern machine

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

3, 5, 7, 9, 11, 13, …aₙ = 3 + 2(n − 1)3n = 15n = 2+27n = 3+29n = 4+211n = 5+213n = 6+2SillyChili
the 10th term: a₁₀ = 21
Each step
Terms shown
6
Try this
  • With Add the same, set d = 2. What do the tops of the bars line up on?
  • Make d negative. What happens to the terms?
  • Switch to Multiply by the same with r = 2. How fast do the bars grow compared with adding?
  • Try r = 0.5. What happens now?
  • Can you predict the 10th term before you read it below the picture?

3Explain

35793, 5, 7, 9, …SillyChili
Step 1 of 5

A sequence is a list in order

A sequence is a list of numbers that follows a rule. Each number is a term. Here each picture has 2 more tiles than the one before: 3, 5, 7, 9, …

Arithmetic and geometric sequencesarithmetic: an=a1+(n−1) dgeometric: an=a1⋅r n−1\begin{aligned} \text{arithmetic: } a_n &= a_1 + (n-1)\,d \\ \text{geometric: } a_n &= a_1 \cdot r^{\,n-1} \end{aligned}

Arithmetic sequences add the same difference and grow along a line. Geometric sequences multiply by the same ratio and grow exponentially.

4Practice

Question 1 of 5

What comes next? 4, 9, 14, 19, …

5Recap

  • A sequence is an ordered list; each number is a term.
  • Arithmetic: add d each time; aₙ = a₁ + (n − 1)d. Its graph is a straight line.
  • Geometric: multiply by r each time; aₙ = a₁ · rⁿ⁻¹. It grows (or shrinks) exponentially.
  • Doubling soon beats adding.