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Mathematics, four units · Ages 16–17

Infinite Geometric Sequences and Convergence Worksheet

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Determine whether each infinite geometric sequence converges to a finite sum. If it does, calculate the sum.

  1. Sequence: 16, 8, 4, 2, ... Find the common ratio q and state if the infinite sum exists.

  2. Calculate the infinite sum for the sequence: 16, 8, 4, 2, ...

  3. Sequence: 5, 15, 45, 135, ... Does an infinite sum exist? Explain why or why not.

  4. Sequence: 12, -6, 3, -1.5, ... Find q and calculate the infinite sum if it exists.

  5. A geometric sequence has a1 = 20 and a3 = 5 with all positive terms. Find q and compute the infinite sum.

  6. A patient takes 100 mg of medicine on day one. Each subsequent day, only 40 percent of the previous day's dose remains active in the body. If daily doses continue indefinitely, what is the cumulative effective total in the body over time?

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Infinite Geometric Sequences and Convergence Worksheet

Answer key

  1. Sequence: 16, 8, 4, 2, ... Find the common ratio q and state if the infinite sum exists.

    q = 8 / 16 = 1/2. Because |1/2| < 1, the infinite sum exists.

  2. Calculate the infinite sum for the sequence: 16, 8, 4, 2, ...

    S = 16 / (1 - 1/2) = 16 / (1/2) = 32.

  3. Sequence: 5, 15, 45, 135, ... Does an infinite sum exist? Explain why or why not.

    No. The ratio q = 15 / 5 = 3. Because |3| >= 1, the series diverges and has no finite sum.

  4. Sequence: 12, -6, 3, -1.5, ... Find q and calculate the infinite sum if it exists.

    q = -6 / 12 = -1/2. Since |-1/2| < 1, S = 12 / (1 - (-1/2)) = 12 / 1.5 = 8.

  5. A geometric sequence has a1 = 20 and a3 = 5 with all positive terms. Find q and compute the infinite sum.

    q^2 = 5 / 20 = 1/4, so q = 1/2. S = 20 / (1 - 1/2) = 40.

  6. A patient takes 100 mg of medicine on day one. Each subsequent day, only 40 percent of the previous day's dose remains active in the body. If daily doses continue indefinitely, what is the cumulative effective total in the body over time?

    a1 = 100 and q = 0.4. Because |0.4| < 1, S = 100 / (1 - 0.4) = 100 / 0.6 = 500/3 or approximately 166.67 mg.