Chapter 2 · Sequences · Concept 2
2-9 Sum of Geometric Sequence (1)
Adding an arithmetic sequence had a beautiful trick: write it backwards and pair it up. Backwards is no use here — the terms are miles apart at one end and tiny at the other. Geometric sequences need a different trick, and it is just as good.
Guiding question
How do you add a list that multiplies?
Learning outcome. Calculate the sum of a geometric sequence of n terms.
The sum of a geometric sequence
With first term a, common ratio r and n terms — and r ≠ 1:
easier when r is less than 1
Sₙ = a(1 − rⁿ)/(1 − r)
easier when r is more than 1
Sₙ = a(rⁿ − 1)/(r − 1)
These are not two formulas. They are one.
Flip the sign of the top and the bottom of a fraction and nothing changes: (1 − rⁿ)/(1 − r) and (rⁿ − 1)/(r − 1) are the same number, always. You may use either one for any question. Choosing the one that keeps the minus signs out of your working is just kindness to yourself.
Write the sum out. Then multiply the whole thing by r and write that underneath. Because multiplying by r turns each term into the next one, the second row is the first row slid along by one — and when you subtract, almost everything meets its twin.
| S = | 4 | 12 | 36 | 108 | 324 |
1(b) First term 8, common ratio 1/2, six terms. Find the sum.
- 1
Pick the version that suits the ratio.
r = 1/2 is less than 1, so use a(1 − rⁿ)/(1 − r)That way the top and bottom both stay positive, and no minus signs get lost.
Work with a partner. Before either of you starts, both say which version of the formula you are using and why. If you pick different ones, do both — you should land on the same number.
1. Find the sum S.
2. Find Sₙ, the sum to the n-th term.
A water treatment system has filters in a row. Each stage traps a third of what the stage before it did, so the amounts removed run 3, 1, 1/3, 1/9, … mg/L. What is the total removed by a four-stage system?
The formula says r ≠ 1. What actually goes wrong at r = 1, and what is the sum of such a sequence really?
Put r = 1 into the bottom of the fraction.
Subtracting a negative ratio wrongly.
With r = −3 the bottom is 1 − (−3) = 4, not −2. Write the bracket in before you simplify it.
Losing the brackets on the power.
(−3)ⁿ is not −3ⁿ, and (1/2)⁶ is not 1/2⁶. Without brackets the power lands on the wrong thing.
Leaving a fraction on the bottom.
Dividing by 1/2 doubles; dividing by 2/3 multiplies by 3/2. Clear it in one deliberate step rather than in your head.
Using n where the power should be n.
The power inside the formula is n, the number of TERMS — not n − 1. That one belongs to the general term, not to the sum.
Turning an exact answer into a decimal.
63/4 is the answer. 15.75 is the same number but the book asks for the fraction, and rounding a longer one loses marks.
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