Key takeaway: A geometric sequence is a list of numbers in which every term is the previous term multiplied by the same constant, called the common ratio.
Grade 9 – Algebra 1F-IF.A.3Functions and Graphs
Definition of Geometric Sequence
In a geometric sequence, neighbors are linked by multiplication instead of addition. The list 3, 12, 48, 192 multiplies by 4 each time, and the list 800, 400, 200, 100 multiplies by one half. The fixed multiplier is the common ratio \(r\).
Growth is much faster than for an arithmetic list when \(r > 1\), and the terms shrink toward zero when \(0 < r < 1\). A negative ratio makes the terms alternate in sign. The graph is a set of dots that lie on an exponential curve, not on a line. Doubling bacteria, a bouncing ball that loses part of its height each bounce, and a phone chain message passed along are all modeled this way.
Formula and notation
Every geometric sequence is described by its first term \(a_1\) and its common ratio \(r\). The term in position \(n\) is
\[a_n = a_1 \cdot r^{\,n-1}\]
Find \(r\) by dividing a term by the one before it: \(r = \frac{a_{n+1}}{a_n}\). The ratio must be the same for every neighboring pair.
Key facts
- To test for a geometric sequence, divide neighboring terms; the quotient must never change.
- If r is between 0 and 1, the terms decrease toward 0.
- A negative ratio makes the signs alternate: +, -, +, -.
- The exponent is n minus 1, because the first term has had no multiplications yet.
Geometric Sequence example
A sequence begins 3, 12, 48, 192. Find the common ratio and the 8th term.
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Ratio: \(\frac{12}{3} = 4\), \(\frac{48}{12} = 4\) and \(\frac{192}{48} = 4\), so \(r = 4\).
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Write the rule: \(a_n = 3 \cdot 4^{\,n-1}\).
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For \(n = 8\): \(a_8 = 3 \cdot 4^{7} = 3 \cdot 16{,}384 = 49{,}152\).
The ratio is 4, and the 8th term is 49,152. By comparison, an arithmetic list with the same first two terms would reach only 3 + 7(9) = 66 at that position.
Common mistake
Watch out
Dev computes the 8th term as \(3 \cdot 4^{8} = 196{,}608\).
Why it is wrong
He used the position number as the exponent. The first term 3 has not been multiplied at all, so reaching term 8 takes 7 multiplications. The exponent is \(n – 1 = 7\), giving \(3 \cdot 4^{7} = 49{,}152\). A check on \(n = 1\) catches this: his rule would give 12 for the first term.
Related terms and lessons
- Arithmetic Sequence
- Exponential Function
- Explicit Formula
- Recursive Formula
- Geometric Series
- Exponential Growth
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