Add Terms Rather than Report the Last Term

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find the sum of the first three terms of a geometric sequence starting at 2 with ratio 2. Candidate answers: (A) 13 (B) 14 (C) 15 (D) 28.
• Independent check inputs: Find the sum of the first three terms of a geometric sequence starting at 3 with ratio 2. Candidate answers: (A) 20 (B) 21 (C) 22 (D) 42.

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 geometric sum adds several values whose consecutive ratio is fixed. Explicitly listing a short supplied sequence distinguishes its total from its last term and from its number of terms.

Here is the method on the supplied teaching case. The three terms are 2,4,8. Their sum is 14. Using a(1−r³)/(1−r) gives the same result when r=2.

A geometric term formula alone does not give a sum. Check whether the question asks for one term or cumulative total; for a finite sum formula, the special case ratio one needs separate handling.

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

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Take it further

Changed-input task: Find the sum of the first three terms of a geometric sequence starting at 9 with ratio 2. Candidate answers: (A) 126 (B) 62 (C) 63 (D) 64. Show the mathematical reason, not only the selected answer. Expected result: 63. The three terms are 9,18,36. Their sum is 63. Using a(1−r³)/(1−r) gives the same result when r=2.

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