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Grade 9 · lesson 3 of 4

Distinguish a Term Number from a Period Count

A geometric sequence multiplies each term by its common ratio.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A geometric sequence starts with a₁=2 and ratio 2. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9.
  • Independent check inputs: A geometric sequence starts with a₁=3 and ratio 2. Find a_4. Candidate answers: (A) 23 (B) 24 (C) 25 (D) 48.

Learn the idea

An exponential pattern changes by a constant multiplicative ratio over equal input steps. This differs from a linear pattern’s constant addition and can model supplied growth, decay or ordered geometric sequence terms. A geometric sequence multiplies each term by its common ratio. Starting from the first term, reaching term n requires n−1 repeated multiplications, which explains its explicit formula.

Here is the method on the supplied teaching case. From term 1 to term 3 there are 2 multiplications by 2. Use a_n=2·2^(n−1) to obtain 8.

Sequence indexing begins at the stated first term here. An exponent n would count one extra multiplication; write the first few terms to check the formula and avoid an off-by-one error.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A geometric sequence starts with a₁=2 and ratio 2. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9. Solution: From term 1 to term 3 there are 2 multiplications by 2. Use a_n=2·2^(n−1) to obtain 8. The final answer is 8. Sequence indexing begins at the stated first term here. An exponent n would count one extra multiplication; write the first few terms to check the formula and avoid an off-by-one error.

Your turn

Check your reasoning

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Helpful tips
  • A geometric sequence multiplies each term by its common ratio.
  • Sequence indexing begins at the stated first term here.
A geometric sequence starts with a₁=3 and ratio 2. Find a_4.

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See the answer and explanation

24

From term 1 to term 4 there are 3 multiplications by 2. Use a_n=3·2^(n−1) to obtain 24.

Take it further

Changed-input task: A geometric sequence starts with a₁=9 and ratio 2. Find a_10. Candidate answers: (A) 4607 (B) 4608 (C) 4609 (D) 9216. Show the mathematical reason, not only the selected answer. Expected result: 4608. From term 1 to term 10 there are 9 multiplications by 2. Use a_n=9·2^(n−1) to obtain 4608.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A geometric sequence starts with a₁=4 and ratio 2. Find a_5.
Need a hint?
  • A geometric sequence multiplies each term by its common ratio.
  • Sequence indexing begins at the stated first term here.
2. A geometric sequence starts with a₁=5 and ratio 2. Find a_6.
Need a hint?
  • A geometric sequence multiplies each term by its common ratio.
  • Sequence indexing begins at the stated first term here.
3. A geometric sequence starts with a₁=6 and ratio 2. Find a_7.
Need a hint?
  • A geometric sequence multiplies each term by its common ratio.
  • Sequence indexing begins at the stated first term here.

Your choices are temporary. Feedback appears after you check all the answers.

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Check the subject sources

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