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 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 ....... a_n are in G.P then a₁ a_n = a₂ a_n-1 = a₃ a_n-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 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 x₁, x₂ , x₃ ...... 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 G₁ , G₂ ------ G_n are ‘n’ geometric means between them, then G₁ , G₂ , G₃ ......... 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}}