Geometric Sequences and Series

Geometric Sequences A geometric sequenceA sequence of numbers where each successive number is the product of the previous number and some constant r., or geometric progressionUsed when referring to a geometric sequence., is a sequence of numbers where each successive number is the product of the previous number and some constant r. an=ran−1 Geometic Sequence And because anan−1=r, the constant factor r is called the common ratioThe constant r that is obtained from dividing any two successive terms of a geometric sequence; anan−1=r.. For example, the following is a geometric sequence, 9,27,81,243,729… Here a1=9 and the ratio between any two successive terms is 3. We can construct the general term an=3an−1 where, a1=9a2=3a1=3(9)=27a3=3a2=3(27)=81a4=3a3=3(81)=243a5=3a4=3(243)=729⋮ In general, given the first term a1 and the common ratio r of a geometric sequence we can write the following: a2=ra1a3=ra2=r(a1r)=a1r2a4=ra3=r(a1r2)=a1r3a5=ra3=r(a1r3)=a1r4⋮ From this we see that any geometric s...