## Sequences and Series

# Geometric Progression (G.P.)

**Tips :**

- If ‘a’ is the first term and r is the common ratio of a finite G.P. Consisting of ‘m’ terms, then the n
^{th}term from the end is given by ar^{m−n } - The n
^{th}term from the end of a G.P with last term l and common ratio ‘r’ is \tt \frac{1}{r^{n-1}} - Four numbers of G.P can be taken as \tt \frac{a}{r^{3}},\frac{a}{r},ar, ar^{3}
- ‘3’ numbers a, b, c are in G.P if and only if \tt \frac{b}{a}=\frac{c}{b}
- Sum of ‘n’ terms of a G.P \tt S_{n} =\frac{a\left(r^{n}-1\right)}{\left(r-1\right)}=\frac{a\left(1-r^{n}\right)}{\left(1-r\right)}
- Sum of an infinite G.P when \tt \mid r\mid <1 \ is \ S_{\propto}=\frac{a}{1-r}\left[-1<r<1\right]
- If r ≥ 1 then S
_{∝}does not exist. - If a
_{1}, a_{2}, a_{3}..... are in G.P then a_{1}k, a_{2}k, a_{3}k ....... are also in G.P - If a
_{1}, a_{2}, a_{3}...... are in G.P then \tt \frac{1}{a_{1}},\frac{1}{a_{2}},\frac{1}{a_{3}} ...... are also in G.P - If a
_{1}, a_{2}, a_{3}....... are in G.P then \tt a_1^k,a_2^k,a_3^k ......... 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}b_{1}, a_{2}b_{2}, a_{3}b_{3}........ are also in G.P \tt \frac{a_{1}}{b_{1}},\frac{a_{2}}{b_{2}},\frac{a_{3}}{b_{3}} ........ are also in G.P - If a
_{1}, a_{2}, a_{3}....... a_{n}are in G.P then a_{1}a_{n}= a_{2}a_{n-1}= a_{3}a_{n-2}= ....... \tt a_{r}=\sqrt{a_{r-k}.a_{r+k}}, o\leq k\leq n-1 - If a
_{1}, a_{2}, a_{3}.......... is a G.P of positive terms, then log a_{1}, log a_{2}, log a_{3}............ is also an A.p and Vice-Versa. - If first term of a G.P of ‘n’ terms is ‘a’ and last term is l, then the product of all terms of the G.P is (al)
^{π/2} - If there be ‘n’ quantities in G.P whose common ratio is r and S
_{m}denotes the sum of the first ‘m’ terms, then the sum of their product taken two by two is \tt \frac{r}{r+1}.S_{n}.S_{n-1} - If a^{x_{1}}, a^{x_{2}}, a^{x_{3}}...... a^{x_{n}}, are in G.P then x
_{1}, x_{2}, x_{3}...... x_{n}will be in A.P - Product of a G.P

**Case 1 :**If number of terms is odd. Then product of terms = (middle term)^{Numbers of terms}

**Case 2 :**If number of terms is even, then product of terms = (Geometric mean of middle terms)^{Numbers of terms} - Geometric mean of ‘a’ and ‘b’ is \tt \sqrt{ab}.
- If ‘n’ number of geometric means are inserted between ‘a’ and ‘b’ then \tt r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}.

**Tricks :**

- The product of ‘n’ geometric means between two given numbers is n
^{th}power of the single G.M between them i.e if ‘a’ and ‘b’ are two given numbers and G_{1}, G_{2}------ G_{n}are ‘n’ geometric means between them, then G_{1}, G_{2}, G_{3}......... G_{n}= \tt \left(\sqrt{ab}\right)^{n} - If R = 0.bbb...... = \tt 0. \overline{b}then \tt R=\frac{b}{10^{1}-1}
- If R = 0. ab ab ab........ = \tt 0. \overline{ab}\ then \ R=\frac{ab}{10^{2}-1}
- If R = 0.abc abc abc ........ then \tt R=\frac{abc}{10^{3}-1}
- If R = \tt R=0.\overline{xy} and ‘x’ denotes the number of digits in 'x' and ‘y’ denotes the number of digits in ‘y’ then \tt R=\frac{xy-x}{10^{x+y}-10^{x}}

### Part1: View the Topic in this video From 42:17 To 52:05

### Part2: View the Topic in this video From 00:40 To 37:42

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1. We know that, a, ar, ar^{2}, ar^{3}, .....ar^{n}^{−1} is a sequence of G.P. Here, the first term is 'a' and the common ratio is 'r'. The general term or n^{th} term of a G.P. is T_{n} = ar^{n}^{−1}.

2. If a be the first term, r the common ratio, then sum S_{n} of first n terms fo a G.P. is given by

\tt S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}, (when |r| < 1)

\tt S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}, (when |r| > 1)

S_{n} = na, (when r = 1)

3. a). When |r| < 1, (or − 1 < r < 1); \tt S_{\infty}=\frac{a}{1-r}

b). If r ≥ 1, S_{∞} then doesn't exist.

4. If a and b are two real numbers then single G.M. between a and b = \tt \sqrt{ab}.

5. Three non-zero numbers a, b, c are in G.P., iff b^{2} = ac.

6. If first term of a G.P. of n terms is a and last term is l, then the product of all terms of the G.p. is (al)^{n/2}.

7. If there be n quantities in G.P. whose common ratio is r and S_{m} denotes the sum of the first m terms, then the sum of their product taken two by two is \frac{r}{r+1}S_{n} \ S_{n-1}.