Geometric Series and the Sum to Infinity
Sum a geometric sequence with the standard formula, and see that when the common ratio is between -1 and 1 the running total settles on a finite limit however many terms are added.
What a learner can do afterwards
- Find the sum of the first 12 terms of a geometric sequence
- Decide from the common ratio whether a sum to infinity exists
- Use a / (1 - r) on a recurring decimal or a repeated-dose model
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Call the common ratio r. Divide a term by its nonzero predecessor to find it, then check other usable pairs: 5, 15, 45 gives r = 3. All-zero terms fit any supplied real ratio, so division cannot determine one uniquely. Alternating signs indicate a negative ratio, as in 4, -8, 16 with r = -2. For nonzero terms, |r| > 1 makes magnitudes grow; |r| < 1 makes them tend to zero. The nth term is first term times r^(n - 1).
For n terms with first term a and ratio r other than 1, the sum is a times (r^n - 1)/(r - 1). Calculate r^n before subtracting 1. If r = 1, every term is a, so the sum is n times a. For 5, 15, 45 over 6 terms, a = 5 and r = 3: 3^6 = 729, so the sum is 5 times (729 - 1)/(3 - 1) = 1820.
For another finite sum, identify the first term a, ratio r and number of terms n separately. The sum is a times (r^n - 1)/(r - 1) when r is not 1. For 4, 8, 16 over 5 terms, a = 4, r = 2 and n = 5. The sum is 4 times (32 - 1)/(2 - 1) = 124. Check by adding: 4 + 8 + 16 + 32 + 64 = 124.
With nonzero first term a, an infinite geometric series has a finite sum exactly when -1 < r < 1. Then r^n tends to zero, leaving a/(1 - r). So 1 + 1/2 + 1/4 + ... totals 1/(1 - 1/2) = 2. If |r| is at least 1 and a is nonzero, the terms do not tend to zero, so the series diverges. If a = 0, every term and partial sum is 0 for any real r.
For a decimal with one recurring digit, separate the place values. Write 0.333... as 0.3 + 0.03 + 0.003 + ... The first term is 0.3 and the ratio is 0.1. Since this ratio is between -1 and 1, use a/(1 - r): 0.3/(1 - 0.1) = 0.3/0.9 = 1/3.
For a decimal with a recurring two-digit block, each block moves two places. Write 0.272727... as 0.27 + 0.0027 + 0.000027 + ... The first term is 0.27 and the ratio is 0.01. Use a/(1 - r): 0.27/(1 - 0.01) = 27/99 = 3/11.
A tank keeps a fifth of its contents each day, then receives 20 new units. Starting empty, just after successive deliveries it holds 20, then 20 + 20(1/5), then 20 + 20(1/5) + 20(1/5)^2. These are partial sums with first term 20 and ratio 1/5. They approach 20/(1 - 1/5) = 25. Any finite starting amount also shrinks toward zero, so the same limiting level applies.
Check the ratio and first term. For finite sums, separate r = 1. With a nonzero first term, an infinite sum exists exactly when -1 < r < 1.
Readings and practice questions are original. Video links are from Khan Academy.
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