Apply a Recursive Rule to the Previous Value

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A recursive sequence has a₁=2 and a_(n+1)=3a_n+1. Find a₂. Candidate answers: (A) 14 (B) 6 (C) 7 (D) 8.
• Independent check inputs: A recursive sequence has a₁=3 and a_(n+1)=3a_n+1. Find a₂. Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9.

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. A recursive definition names a starting term and an operation producing the next term. Substituting the previous term’s actual value follows that definition one step at a time.

Here is the method on the supplied teaching case. Apply the supplied rule once to a₁: a₂=3(2)+1=7. The subscript n marks a term position, not a multiplication by n.

The subscript labels a position rather than multiplication. Do not replace the previous-term value with the term number; a recursion without a starting value would not determine this numerical sequence.

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.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: A recursive sequence has a₁=9 and a_(n+1)=3a_n+1. Find a₂. Candidate answers: (A) 27 (B) 28 (C) 29 (D) 56. Show the mathematical reason, not only the selected answer. Expected result: 28. Apply the supplied rule once to a₁: a₂=3(9)+1=28. The subscript n marks a term position, not a multiplication by n.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.