Geometric Sequences

Pre-Calculus · Axiom Academy

A sequence where each term is the one before it times a fixed number — the common ratio. A geometric sequence is a list of numbers in which the ratio between each term and the one before it is always the same. That constant is the common ratio , written r . Take — every term is the previous one times 2 , so r = 2 . Divide any term by the one before it — you always get r . To reach the n th term you start at a_1 and multiply by r over and over. Getting to a_2 takes one multiplication, a_3 takes two, a_4 takes three — so reaching a_n takes exactly n-1 of them. Counting those steps gives the formula directly. a_1= first term, r= common ratio, n= term number Worked example — the 10th term For we have a_1 = 3 and r = 2 . The exponent is n-1 = 9 : Once you have a starting term, the single number r decides the entire personality of the sequence. Plot the terms and the difference is unmistakable: r bigger than 1 explodes upward, r between 0 and 1 melts toward zero, and a negative r flips the sign at every step. Grows without bound — exponential blow-up (e.g. with r=3 ). Shrinks toward 0 — decay (e.g. with ). Alternates sign each step — the terms zig-zag across zero. Every term equals a_1 — a constant sequence. (And , or all terms after a_1 collapse to 0 .) To recover r , divide any term by the one before it. For : Check the next pairs: and . The ratio holds, so the sequence is geometric. Graph: a straight line. Example: Graph: an exponential curve. Example:

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