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Grade 11 · lesson 1 of 4

Count Equal Additive Steps from the First Term

An arithmetic sequence adds the same common difference between consecutive terms.

About 25 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: An arithmetic sequence has a₁=2, common difference 3. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9.
  • Independent check inputs: An arithmetic sequence has a₁=3, common difference 3. Find a_4. Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24.

Learn the idea

A sequence distinguishes an indexed term from the sum of several terms. Arithmetic and geometric structures support different formulas, while a recursive definition determines each next term from previously defined values. An arithmetic sequence adds the same common difference between consecutive terms. Its nth term equals the first value plus n−1 copies of that difference because index one starts before any step.

Here is the method on the supplied teaching case. Term 3 is 2 steps after term 1. Therefore a_3=2+3(2)=8. The exponent rule for geometric sequences is not used.

The number of steps is one less than the term position. Do not use a geometric exponent for an additive sequence; write early terms to verify indexing and the stated common difference.

  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: An arithmetic sequence has a₁=2, common difference 3. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9. Solution: Term 3 is 2 steps after term 1. Therefore a_3=2+3(2)=8. The exponent rule for geometric sequences is not used. The final answer is 8. The number of steps is one less than the term position. Do not use a geometric exponent for an additive sequence; write early terms to verify indexing and the stated common difference.

Your turn

Check your reasoning

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Helpful tips
  • An arithmetic sequence adds the same common difference between consecutive terms.
  • The number of steps is one less than the term position.
An arithmetic sequence has a₁=3, common difference 3. Find a_4.

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See the answer and explanation

12

Term 4 is 3 steps after term 1. Therefore a_4=3+3(3)=12. The exponent rule for geometric sequences is not used.

Take it further

Changed-input task: An arithmetic sequence has a₁=9, common difference 3. Find a_10. Candidate answers: (A) 35 (B) 36 (C) 37 (D) 72. Show the mathematical reason, not only the selected answer. Expected result: 36. Term 10 is 9 steps after term 1. Therefore a_10=9+3(9)=36. The exponent rule for geometric sequences is not used.

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More practice: try new cases

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More practice: try new cases
1. An arithmetic sequence has a₁=4, common difference 3. Find a_5.
Need a hint?
  • An arithmetic sequence adds the same common difference between consecutive terms.
  • The number of steps is one less than the term position.
2. An arithmetic sequence has a₁=5, common difference 3. Find a_6.
Need a hint?
  • An arithmetic sequence adds the same common difference between consecutive terms.
  • The number of steps is one less than the term position.
3. An arithmetic sequence has a₁=6, common difference 3. Find a_7.
Need a hint?
  • An arithmetic sequence adds the same common difference between consecutive terms.
  • The number of steps is one less than the term position.

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

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