Mathematics · Ch 11 — Sequences and Series

Geometric Progression

11.3

Geometric Progression

A sequence $t_1, t_2, t_3, \ldots, t_n, \ldots$ is a Geometric Progression (G.P.) if the ratio of any term to the one before it, $\dfrac{t_{n+1}}{t_n}=r$ , is constant for all $n$ ; $r$ is called the common ratio. A G.P. can always be written as $a, ar, ar^2, \ldots$ where $a$ is the first term and $r$ is the common ratio.

Examples: (i) $2,4,8,16,\ldots$ has $a=2,\ r=2$ . (ii) $1,\dfrac13,\dfrac19,\dfrac1{27},\ldots$ has $a=1,\ r=\dfrac13$ . (iii) $1,-1,1,-1,1,-1,\ldots$ has $a=1,\ r=-1$ . …

Misc 1Properties of a Geometric Progression

Worked out. A short list of closure properties that follow directly from the G.P. definition and are used repeatedly in later problems. If $t_1, t_2, t_3, \ldots, t_n$ are in G.P., then: (i) their reciprocals $\dfrac1{t_1}, \dfrac1{t_2}, \dfrac1{t_3}, \ldots, \dfrac1{t_n}$ are also in G.P. (with ratio $1/r$ ); (ii) multiplying every term by the same nonzero constant $k$ , i.e. $kt_1, kt_2, kt_3, \ldots, kt_n$ , keeps the sequence a G.P. (ratio unchanged); and (iii) raising every term to the same power $n$ , i.e. $t_1^{\,n}, t_2^{\,n}, t_3^{\,n}, \ldots$ , is also a G.P. (with ratio $r^n$ ). These are the standard toolkit for quickly transformin …

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