Geometric Series and the Sum to Infinity · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Geometric sums that settle down

Mathematics · Algebra · ages 17-18
Name ______________________   Date ____________

  1. What is the sum of the first 5 terms of 1, 3, 9, ...?

    • 81
    • 243
    • 121
    • 120

  2. The sequence 3, 6, 12, ... is geometric. What is the common ratio?

    • 2
    • 3
    • 6

  3. What is the sum of the first 6 terms of 2, 4, 8, ...?

    Answer: ______________

  4. For geometric series with nonzero first terms, which common ratio gives a finite sum to infinity?

    • Common ratio 2
    • Common ratio 1
    • Common ratio 1/2

  5. Write the recurring decimal 0.555... as a fraction in lowest terms.

    • 1/2
    • 5/9
    • 5/10
    • 11/20

  6. The sequence 2, 6, 18, ... is geometric. What is the sum of its first 8 terms?

    Answer: ______________

  7. A geometric series with a nonzero first term and common ratio -3/2 settles on a finite total. True or false?

    Circle one:   True   False

  8. The sequence 3, 6, 12, ... is geometric. What is the sum of its first 12 terms?

    Answer: ______________

  9. Write the recurring decimal 0.454545... as a fraction in lowest terms.

    • 5/11
    • 45/100
    • 1/2

  10. A patient takes a 40 mg dose each week, and 1/5 of the drug in the body is still there a week later. What level in mg does the amount just after a dose settle at?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Geometric sums that settle down W1-mt_HHxFOv6vXS-s1

  1. 121 · Ratio 3, so the sum is (3^5 - 1)/2 = 242/2 = 121. 81 is the 5th term alone.
  2. 2 · Divide any term by the one before it: 6/3 = 2. The ratio is 2.
  3. 126 · The ratio is 2, so the sum is 2 times (2^6 - 1). That is 2 times 63 = 126.
  4. Common ratio 1/2 · With a nonzero first term, convergence requires -1 < r < 1. Only 1/2 satisfies this condition; ratios 2 and 1 give divergent series.
  5. 5/9 · 0.5 + 0.05 + 0.005 + ... has first term 0.5 and ratio 0.1, totalling 0.5/0.9 = 5/9.
  6. 6560 · First term 2 and ratio 3 give 2 times (3^8 - 1)/(3 - 1) = 6561 - 1 = 6560.
  7. False · The absolute value of -3/2 is 3/2, above 1. With a nonzero first term, the terms do not tend to zero, so the series diverges.
  8. 12285 · With first term 3 and ratio 2, the sum is 3 times (2^12 - 1). That is 3 times 4095 = 12285.
  9. 5/11 · It splits into 0.45 + 0.0045 and so on with ratio 0.01, totalling 0.45/0.99. That simplifies to 5/11.
  10. 50 · Just after a dose the amount is 40 + 40(1/5) + 40(1/5)^2 + ..., which totals 40/(1 - 1/5) = 50.
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