Mathematics, four units · Grade 11

When does an infinite geometric sequence have a finite sum?

Only when the absolute value of the common ratio is less than one, that is when the ratio lies strictly between minus one and one and is not zero. Then the terms shrink fast enough and the sum converges to the first term over one minus the ratio. If the absolute value reaches one or more, the sum grows without bound.

Learning objectives

Multiplying instead of adding

In a geometric sequence each term comes from the one before by multiplying by the same number, the common ratio, written q. You find q by dividing a term by its predecessor — not by subtracting.

The general term is the first term times q raised to the number of steps. Again, reaching the nth term takes n minus one steps, exactly as in an arithmetic sequence.

When the two given terms are not adjacent, dividing one by the other gives q raised to the number of steps between them. Take the appropriate root — and note that an even power has two roots, positive and negative, and sometimes both are valid.

The sum of finitely many terms

The sum of the first n terms is the first term times q to the power n minus one, all over q minus one. That form suits a ratio above one; for a ratio below one it is tidier to flip both signs and work with positive numbers.

When the ratio equals one the formula divides by zero and is meaningless. In that case all the terms are equal, and the sum is simply the term times the number of terms.

In an exam it is worth a sanity check on the order of magnitude. A sequence with q equal to 2 and ten terms sums to roughly a thousand times its first term; an answer far smaller than that signals a slip in the power.

The infinite sum

Adding infinitely many terms raises one question: do they shrink fast enough? The condition is that the absolute value of q is below one. Then, and only then, the sum converges, to the first term over one minus q.

The intuition: a ball that rebounds to two thirds of its previous height each time travels a perfectly finite distance, even though it bounces infinitely often. The bounces become so short that their total is bounded.

If the ratio is negative but smaller than one in absolute value — minus a half, say — the sum still exists. The terms alternate in sign and shrink, and the formula works as usual with the negative ratio.

Worked examples

  1. In a geometric sequence the first term is 5 and the third is 45. Find the ratio.

    1. Between positions one and three there are two steps
    2. 45 over 5 is 9, and that is q squared
    3. Take the square root

    Answer: q is 3 or q is minus 3 — two possible sequences

  2. What is the sum of the first 6 terms of 2, 6, 18 and so on?

    1. The ratio: 6 over 2, which is 3
    2. 3 to the power 6 is 729
    3. Substitute: 2 times (729 minus 1) over (3 minus 1)

    Answer: 2 times 728 over 2, which is 728

  3. A ball is dropped from 3 metres and rebounds to two thirds of its height each time. What is the total of the rebound heights?

    1. The first rebound reaches 3 times two thirds, which is 2
    2. The ratio is two thirds, whose absolute value is below one
    3. Substitute: 2 over (1 minus two thirds)

    Answer: 2 over one third, which is 6 metres

Common mistakes

Finding the ratio by subtracting instead of dividing
Differences belong to arithmetic sequences and ratios to geometric ones. Subtracting in a geometric sequence gives a number that changes at every step and is not the ratio.
Using the infinite sum formula when the ratio exceeds one
The formula will return a number, and that number will be negative or meaningless. The sum simply does not exist, and the correct answer is to say so and cite the convergence condition.
Automatically rejecting a negative ratio
A negative ratio is perfectly valid and produces a sequence alternating in sign. An infinite sum exists for it too, as long as its absolute value is below one.

What to remember

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