Hello MyRankers, Here is the explanation of Geometric Progression……

A sequence of non-zero numbers is called a geometric progression (abbreviated as G.P.). If the ratio of a term and the term preceding to it is always a constant quantity.

The constant ratio is called the common ratio of the G.P.

In other words, a sequence, a₁, a₂, a₃, a₄ … a_{n} … is called a geometric progression \(\frac{{{a}_{n+1}}}{{{a}_{n}}}\) = Constant for all n ϵ N.

**GEOMETRIC SERIES: **If a₁, a₂, a₃ … a_{n} … are in G.P, then the expression a₁ + a₂ + a₃ + … a_{n} + … is called a geometric series.

**SELECTION OF TERMS IN G.P: **Sometimes it is required to select a finite number of terms in G.P. It is always convenient if we select the terms in the following manner:

No. of terms |
Terms | Common Ratio |

3 | , a, ar |
r |

4 |
,, ar,ar³ | r² |

5 | , , a, ar, ar² |
r |

If the product of the numbers is not given, then the numbers are taken as a, ar, ar², ar³, …

**Properties of Geometric Progressions:**

**PROPERTY I:** If all the terms of a G.P .are to be multiplied or divided by the same non-zero constant, then it remains a G.P. with the same common ratio.

**PROPERTY II: **The reciprocals of the terms of a given G.P. form a G.P.

**PROPERTY III: **If each term of a G.P. is raised to the same power, the resulting sequence also forms a G.P.

**PROPERTY IV: **In a finite G.P. the product of the terms equidistant from the beginning and the end is always same and is equal to the product of the first and last term.

**PROPERTY V: **Three non-zero numbers a, b, c are in G.P. if b² = ac.

**PROPERTY VI: **If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P.

**PROPERTY VII: **If a₁, a₂, a₃, a₄ … a_{n} … be a G.P of non-zero non-negative terms, then log a₁, log a₂, … loga_{n}, … is an A.P. and viceversa.

**SUM OF n TERMS OF A G.P: **The sum of n terms of a G.P. with first term ‘a’ and common ratio ‘r’ is given by \({{S}_{n}}=a\left( \frac{{{r}^{n}}-1}{r-1} \right)\,\) or \({{S}_{n}}=a\left( \frac{1-{{r}^{n}}}{1-r} \right)\,\), r ≠ 1.

If l* *is the last term of the G.P., then l = arⁿ¯¹.

\({{S}_{n}}=a\left( \frac{1-{{r}^{n}}}{1-r} \right)\,=\frac{a-a{{r}^{n}}}{1-r}=\frac{a-\left( a{{r}^{n-1}} \right)r}{1-r}=\frac{a-lr}{r-1}\).

Thus, \({{S}_{n}}=\frac{a-lr}{1-r}\) or \(\frac{lr-a}{r-1}\), r ≠ 1.

If n geometric means are inserted between two quantities, then the product of n geometric means is the nth power of the single geometric mean between the two quantities.

**RELATION BETWEEN ARITHMETIC MEAN AND GEOMETRIC MEAN:**

**PROPERTY I: **If A and G are respectively arithmetic and geometric means between two positive numbers a and b, then A > G.

**PROPERTY II: **If A and G are respectively arithmetic and geometric means between two positive quantities a and b, then the quadratic equation having a, b as its roots is X² – 2AX + G² = 0.

**PROPERTY III: **If A and G be the A.M and G.M. between two positive numbers, then the numbers are\(A\pm \sqrt{{{A}^{2}}+{{G}^{2}}}\).