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
1 · Read
A geometric sequence multiplies by the same ratio every step, which outruns anything adding a fixed amount. Find the ratio by dividing any term by the one before it, and check it works everywhere: 3, 6, 12 has ratio 2. Alternating signs just mean a negative ratio. The nth term is the first term times the ratio to the power n - 1, so check n = 1 gives the first term back.
Add finite runs with first term times (r^n - 1)/(r - 1). For 3, 6, 12 over 12 terms: 2^12 = 4096, so 3 times 4095 = 12285. For 2, 4, 8 over 6 terms: 2 times (64 - 1) = 126.
An endless sum lands on a finite total only when its terms shrink fast, which happens exactly when the ratio sits strictly between -1 and 1. Then the r^n part melts to zero and first term/(1 - r) is left. So 1 + 1/2 + 1/4 + ... totals 1/(1 - 1/2) = 2. With a ratio of 2 or 1 the terms never shrink, so no finite total exists.
Recurring decimals are this formula in disguise. Write 0.777... as 0.7 + 0.07 + 0.007 + ..., a series with first term 0.7 and ratio 0.1, totalling 0.7/0.9 = 7/9. The same moves turn 0.454545... into 0.45 + 0.0045 + ..., with ratio 0.01, totalling 0.45/0.99 = 45/99 = 5/11. A tank getting 12 new units daily while keeping a quarter of its total settles at 12/(1 - 1/4) = 16.
Find the ratio first, use it for finite sums, and trust first term/(1 - r) only when the ratio sits between -1 and 1.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.