Sequence and Series
3.0 Geometric Sequence or Geometric Progression (G.P.)
3.0 Geometric Sequence or Geometric Progression (G.P.)
It is a sequence in which the ratio of any term and its just preceding term is constant throughout. This constant is called the common ratio $(r)$ of the G.P.
In a G.P. first term of the sequence is non-zero and each of the term is obtained by multiplying its just preceding term by a constant $r$.
If sequence $\{ {t_1},{t_2},{t_3},...\} $ is such that $\frac{{{t_n}}}{{{t_{n - 1}}}} = {\text{constant = }}r{\text{ }}\forall n \in N$, then it is a G.P.
How to represent G.P.
First term ${a_1}$, Common ratio $=r$
${a_1},{a_2},{a_3},...,{a_n}$ are in G.P. then, $$\frac{{{a_2}}}{{{a_1}}} = \frac{{{a_3}}}{{{a_2}}} = ... = \frac{{{a_n}}}{{{a_{n - 1}}}} = {\text{constant = }}r$$
G.P is represented in terms of first term $a$ and common ratio $r$ as $$a,ar,a{r^2},a{r^3},a{r^4},...,a{r^{n - 1}}$$
- $n$th term of G.P. i.e., $${T_n} = a{r^{n - 1}}$$ where $r = \frac{{{T_n}}}{{{T_{n - 1}}}}$
- $n$th term of G.P. from last i.e., $${T'_n} = \frac{l}{{{r^{n - 1}}}}$$ where $l$ is the last term.
- Sum of first $n$ terms of G.P. i.e., \[{S_n} = \left\{ {\begin{array}{c} {\frac{{a({r^n} - 1)}}{{(r - 1)}} = \frac{{lr - a}}{{r - 1}},{\text{ if }}r \ne 1} \\ {na,{\text{ if }}r = 1} \end{array}} \right.\]
Proof: $$\begin{equation} \begin{aligned} {S_n} = a + ar + a{r^2} + a{r^3} + ... + a{r^{n - 1}}...(1) \\ {S_n} = {\text{ }}a{\text{ }} + ar{\text{ }} + a{r^2} + ... + a{r^{n - 2}} + a{r^{n - 1}}...(2) \\\end{aligned} \end{equation} $$
Multiply equation $(2)$ by $r$, we get $$r{S_n} = {\text{ }}ar{\text{ }} + a{r^2}{\text{ }} + a{r^3} + ... + a{r^{n - 1}} + a{r^n}...(3)$$
Subtracting $(1)$ from $(3)$, we get $$\begin{equation} \begin{aligned} {S_n}(1 - r) = a + ar - ar + a{r^2} - a{r^2} + ... + a{r^n} \\ {S_n}(1 - r) = a - a{r^n} \\ {S_n} = \frac{{a - a{r^n}}}{{(1 - r)}} = \frac{{a(1 - {r^n})}}{{(1 - r)}} \\\end{aligned} \end{equation} $$
When $r > 1$, $${S_n} = \frac{{a({r^n} - 1)}}{{(r - 1)}}$$
When $r < 1$, $${S_n} = \frac{{a(1 - {r^n})}}{{(1 - r)}}$$
Now, last term $$\begin{equation} \begin{aligned} l = a{r^{n - 1}} \\ l = \frac{{a{r^n}}}{r} \\ {r^n} = \frac{{lr}}{a} \\\end{aligned} \end{equation} $$
Put the value in $${S_n} = \frac{{a({r^n} - 1)}}{{(r - 1)}}$$ we get $${S_n} = \frac{{lr - a}}{{r - 1}}$$
- Sum of an infinite G.P. when $\left| r \right| < 1$ is $${S_n} = \frac{a}{{1 - r}}$$ Note: When $\left| r \right| \geqslant 1$, the series is divergent and its sum is not possible.
Note:
- If a sequence is in G.P., and each term of the sequence is multiplied or divided by any constant term, say $k$, $(k \ne 0)$ then the resulting series will also be in G.P.
- If ${a_1},{a_2},{a_3},...$ are in G.P., then $\frac{1}{{{a_1}}},\frac{1}{{{a_2}}},\frac{1}{{{a_3}}},...$ are also in G.P.
- If ${a_1},{a_2},{a_3},...$ and ${b_1},{b_2},{b_3},...$ are two G.P.s, then ${a_1} \pm {b_1},{a_2} \pm {b_2},{a_3} \pm {b_3},...$ are not in G.P.
- If ${a_1},{a_2},{a_3},...$ and ${b_1},{b_2},{b_3},...$ are two G.P.s, then ${a_1}{b_1},{a_2}{b_2},{a_3}{b_3},...$ and ${a_1}/{b_1},{a_2}/{b_2},{a_3}/{b_3},...$ are also in G.P.
- If ${a_1},{a_2},{a_3},...{a_n}$ is in G.P., then ${a_1}^r,{a_2}^r,{a_3}^r,...,{a_n}^r$ is also in G.P. when $r \in Q$.
- If ${a_1},{a_2},{a_3},...{a_n}$ is in G.P., then ${a_1}{a_n} = {a_2}{a_{n - 1}} = {a_3}{a_{n - 2}} = ...$.
- $2m+1$ $(m \in N)$ terms in G.P. can be taken as $$\frac{a}{{{r^m}}},\frac{a}{{{r^{m - 1}}}},...,\frac{a}{r},a,ar,...,a{r^{m - 1}},a{r^m}$$ For example: Three terms in G.P. can be taken as $$\frac{a}{r},a,ar$$ Five terms in G.P. can be taken as $$\frac{a}{{{r^2}}},\frac{a}{r},a,ar,a{r^2}$$
- $2m$ $(m \in N)$ terms in G.P. can be taken as $$\frac{a}{{{r^{2m - 1}}}},\frac{a}{{{r^{2m - 3}}}},...,\frac{a}{{{r^3}}},\frac{a}{r},ar,a{r^3},...,a{r^{2m - 3}},a{r^{2m - 1}}$$ For example: Four terms in G.P. can be taken as $$\frac{a}{{{r^3}}},\frac{a}{r},ar,a{r^3}$$ Six terms in G.P. can be taken as $$\frac{a}{{{r^5}}},\frac{a}{{{r^3}}},\frac{a}{r},ar,a{r^3},a{r^5}$$
