 # 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 nth term from the end is given by arm−n
• The nth 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 a1, a2 , a3 ..... are in G.P then a1 k, a2 k, a3 k ....... are also in G.P
• If a1, a2 , a3 ...... are in G.P then \tt \frac{1}{a_{1}},\frac{1}{a_{2}},\frac{1}{a_{3}} ...... are also in G.P
• If a1, a2 , a3 ....... are in G.P then \tt a_1^k,a_2^k,a_3^k ......... are also in G.P
• If a1, a2 , a3 ........ and b1, b2 , b3 ......... are two G.P s then a1 b1 , a2 b2 , a3 b3 ........ 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 a1, a2, a3 ....... an are in G.P then a1 an = a2 an-1 = a3 an-2 = ....... \tt a_{r}=\sqrt{a_{r-k}.a_{r+k}}, o\leq k\leq n-1
• If a1, a2 , a3 .......... is a G.P of positive terms, then log a1 , log a2 , log a3 ............ 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 Sm 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 x1, x2 , x3 ...... xn 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 nth power of the single G.M between them i.e if ‘a’ and ‘b’ are two given numbers and G1 , G2 ------ Gn are ‘n’ geometric means between them, then G1 , G2 , G3 ......... Gn = \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}}

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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. We know that, a, ar, ar2, ar3, .....arn−1 is a sequence of G.P. Here, the first term is 'a' and the common ratio is 'r'. The general term or nth term of a G.P. is Tn = arn−1.

2. If a be the first term, r the common ratio, then sum Sn 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)
Sn = 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 b2 = 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 Sm 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}.