Geometric Sequences and Common Ratios
Recognise a sequence built by multiplying by the same number each time, find the common ratio and the nth term rule, and link it to growth and decay.
What a learner can do afterwards
- Find the common ratio by dividing consecutive terms
- Write the nth term as a first term times r to a power
- Continue a decaying sequence with a ratio between 0 and 1
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A geometric sequence multiplies by the same number each step. This is the common ratio r. Divide a term by its nonzero predecessor: in 9, 18, 36, 72, the quotient 18/9 is 2. Check another usable pair. A constant difference instead describes an arithmetic pattern. All-zero terms fit any supplied real ratio, so division cannot determine their ratio uniquely.
To continue a geometric sequence, find the ratio and multiply the last known term by it. In 5, 15, 45, 135, the ratio is 15/5 = 3. The next term is 135 times 3 = 405, followed by 405 times 3 = 1215.
Ratios strictly between 0 and 1 shrink term magnitudes toward zero and model decay. 100, 50, 25 continues to 12.5 then 6.25 with ratio 0.5. Track the term number carefully: mistaking it for the power is the classic error.
For first term a and ratio r, term n is a times r^(n - 1). The first term needs no multiplication, so its power is zero. For first term 4 and ratio 3, term 4 is 4 times 3^3 = 108. Substitute the term number, subtract 1 for the power, then calculate.
An explicit rule gives a term directly; a recursive rule gives the first term and how to get the next. With first term 5 and each next term 3 times the previous term, the explicit rule is 5 times 3^(n - 1). Either rule can be converted to the other. To locate 1215, solve 5 times 3^(n - 1) = 1215. Dividing by 5 gives 243 = 3^5, so n - 1 = 5 and n = 6.
Divide neighbours for the ratio, then first term times ratio to n minus 1.
Readings and practice questions are original. Video links are from Khan Academy.
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