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

Geometric sums that settle down

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

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

    • 126
    • 64
    • 128
    • 62

  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. Which of these geometric series has a sum to infinity?

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

  5. Write the recurring decimal 0.777... as a fraction in lowest terms.

    • 7/9
    • 7/10
    • 3/4
    • 7/8

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

    Answer: ______________

  7. A geometric series with 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. Each day a tank gets 12 new units and keeps a quarter of its total from the day before. What level does it settle at in the long run?

    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. 126 · The ratio is 2, so the sum is 2 times (2 to the 6 minus 1) = 2 times 63 = 126.
  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 · The endless sum exists only for ratios strictly between -1 and 1. Only 1/2 passes.
  5. 7/9 · 0.777... is 0.7 plus 0.07 plus 0.007 and so on, a geometric series with first term 0.7 and ratio 0.1, totalling 0.7 over 0.9 = 7/9.
  6. 12285 · With first term 3 and ratio 2, the sum is 3 times (2 to the 12 minus 1) = 3 times 4095 = 12285.
  7. False · The ratio must sit strictly between -1 and 1. Since -3/2 sits outside, the sentence is false.
  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. 16 · The long run total is a sum to infinity with first term 12 and ratio 1/4: 12 over (1 minus 1/4) = 16.
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