\displaystyle \textbf{1. Geometric Progression (G.P.)}
\displaystyle \text{A sequence of non-zero numbers is called a Geometric Progression (G.P.)}
\displaystyle \text{if the ratio of a term and the term preceding it is always a constant quantity.}
\displaystyle \text{The constant ratio is called the common ratio.}
\\

\displaystyle \textbf{2. Geometric Series}
\displaystyle \text{If }a_1,a_2,a_3,\ldots,a_n,\ldots\text{ is a G.P., then the expression}
\displaystyle a_1+a_2+a_3+\cdots+a_n+\cdots
\displaystyle \text{is called a geometric series.}
\\

\displaystyle \textbf{3. nth Term of a G.P.}
\displaystyle \text{The nth term of a G.P. with first term }a\text{ and common ratio }r\text{ is given by}
\displaystyle a_n=ar^{\,n-1}
\\

\displaystyle \textbf{4. nth Term from the End}
\displaystyle \text{If a G.P. consists of }m\text{ terms, then the nth term from the end is the }(m-n+1)^{th}
\displaystyle \text{term from the beginning and is given by}
\displaystyle ar^{\,m-n}
\displaystyle \text{If }l\text{ is the last term of a G.P., then nth term from the end is}
\displaystyle l\left(\frac{1}{r}\right)^{n-1}
\\

\displaystyle \textbf{5. Product of Equidistant Terms}
\displaystyle \text{In a G.P., the product of the terms equidistant from the beginning and the end is always the same}
\displaystyle \text{and is equal to the product of the first and last terms.}
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\displaystyle \textbf{6. Convenient Selection of Terms in a G.P.}
\displaystyle \text{For }3\text{ terms: }\frac{a}{r},\;a,\;ar\qquad \text{Common ratio }=r
\displaystyle \text{For }4\text{ terms: }\frac{a}{r^3},\;\frac{a}{r},\;ar,\;ar^3\qquad \text{Common ratio }=r^2
\displaystyle \text{For }5\text{ terms: }\frac{a}{r^2},\;\frac{a}{r},\;a,\;ar,\;ar^2\qquad \text{Common ratio }=r
\\

\displaystyle \textbf{7. Sum of n Terms of a G.P.}
\displaystyle \text{If }r\ne 1,\text{ then}
\displaystyle S_n=a\left(\frac{r^n-1}{r-1}\right)
\displaystyle \text{or}
\displaystyle S_n=a\left(\frac{1-r^n}{1-r}\right)
\displaystyle \text{If }r=1,\text{ then}
\displaystyle S_n=na
\displaystyle \text{Also, if }l\text{ is the last term, then}
\displaystyle S_n=\frac{a-lr}{1-r}
\displaystyle \text{or}
\displaystyle S_n=\frac{lr-a}{r-1}
\\

\displaystyle \textbf{8. Multiplication or Division by a Constant}
\displaystyle \text{If all the terms of a G.P. are multiplied or divided by the same non-zero constant,}
\displaystyle \text{it remains a G.P. with the same common ratio.}
\\

\displaystyle \textbf{9. Reciprocals of a G.P.}
\displaystyle \text{The reciprocals of the terms of a given G.P. form a G.P.}
\\

\displaystyle \textbf{10. Powers of Terms of a G.P.}
\displaystyle \text{If each term of a G.P. is raised to the same power, the resulting sequence also forms a G.P.}
\\

\displaystyle \textbf{11. Geometric Mean}
\displaystyle \text{Three numbers }a,b,c\text{ are in G.P. iff}
\displaystyle b^2=ac
\displaystyle \text{If }a,b,c\text{ are in G.P., then }b\text{ is known as the geometric mean of }a\text{ and }c.
\\

\displaystyle \textbf{12. Terms Chosen at Regular Intervals}
\displaystyle \text{If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed}
\displaystyle \text{also forms a G.P.}
\\

\displaystyle \textbf{13. Insertion of Geometric Means}
\displaystyle \text{Let }a\text{ and }b\text{ be two given numbers. If }n\text{ numbers }
\displaystyle G_1,G_2,G_3,\ldots,G_n
\displaystyle \text{are inserted between }a\text{ and }b
\displaystyle \text{such that}
\displaystyle a,\;G_1,\;G_2,\ldots,G_n,\;b
\displaystyle \text{is a G.P., then }G_1,G_2,\ldots,G_n\text{ are known as }n\text{ geometric means between }a\text{ and }b.
\displaystyle \text{The common ratio is given by}
\displaystyle r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}
\\

\displaystyle \textbf{14. Geometric Mean of Two Numbers}
\displaystyle \text{The geometric mean of }a\text{ and }b\text{ is given by}
\displaystyle \sqrt{ab}
\\

\displaystyle \textbf{15. Product of Inserted Geometric Means}
\displaystyle \text{If }n\text{ geometric means are inserted between two quantities, then the product of the }n
\displaystyle \text{geometric means is the }n^{th}\text{ power of the single geometric mean between the two quantities.}
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\displaystyle \textbf{16. Arithmetic Mean and Geometric Mean}
\displaystyle \text{If }A\text{ and }G\text{ are respectively arithmetic and geometric means between two positive numbers}
\displaystyle a\text{ and }b,\text{ then}
\displaystyle \text{(i) }A>G
\displaystyle \text{(ii) The quadratic equation having }a,b\text{ as its roots is}
\displaystyle x^2-2Ax+G^2=0
\displaystyle \text{(iii) }a:b=\left(A+\sqrt{A^2-G^2}\right):\left(A-\sqrt{A^2-G^2}\right)
\\

\displaystyle \textbf{17. Ratio of Numbers from AM and GM}
\displaystyle \text{If AM and GM between two numbers are in the ratio }m:n,\text{ then the numbers are in the ratio}
\displaystyle m+\sqrt{m^2-n^2}:m-\sqrt{m^2-n^2}
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