Common Ratio

Algebra 2 · Axiom Academy

LESSON Geometric Sequences: The Common Ratio One fixed multiplier, called the common ratio, decides everything a geometric sequence does. 1. A Constant Ratio, Not a Constant Difference What makes a sequence geometric ? The jump from each term to the next is a multiplication by the same number , so the ratio of any term to the one before it never changes. Contrast that with an arithmetic sequence, where it is the difference that stays constant. Geometric — the ratio is fixed Arithmetic — the difference is fixed 2. Finding r : Divide Any Term by the One Before It To pin down the common ratio, take any term and divide it by the term immediately before it. Because the ratio is constant, it does not matter which pair you choose — every consecutive pair gives back the same r . Slide from pair to pair and the quotient never budges: To reach the n -th term, start at the first term a_1 and multiply by r over and over. The catch is the count : getting from a_1 to a_n takes exactly n-1 multiplications — so the exponent on r is always one less than the term number. Building 2, 6, 18, 54, 162 from a_1 = 2 , r = 3 Each new term adds one more factor of 3 , and the exponent counts those factors: term 5 needs 5-1 = 4 of them. Here a_1 is the first term, r is the common ratio, and n is the term number you want. 4. What the Common Ratio Controls

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