We will discuss about different types of Geometric Progression formulae.

**1.** The general form of a Geometric Progression is {a, ar, ar\(^{2}\), ar\(^{3}\), ar\(^{4}\), ............}, where ‘a’ and ‘r’ are called the first term and common ratio (abbreviated as C.R.) of the Geometric Progression.

**2.** The general term or nth term of a Geometric Progression with first term ‘a’ and common ratio ‘r’ is given by t\(_{n}\) = a ∙ r\(^{n - 1}\)

**3.** The sum of first n terms of the Geometric Progression whose first term ‘a’ and common ratio ‘r’ is given by

S\(_{n}\) = a(\(\frac{r^{n} - 1}{r - 1}\))

⇒ S\(_{n}\) = a(\(\frac{1 - r^{n}}{1 - r}\)), r ≠ 1.

**4.** If x be the geometric mean of two given quantities a and b then, x = ±√ab

When a and b are two quantities of opposite symbols then, the geometric mean between these quantities does not exist.

**5.** (i) If
the product of three numbers in Geometric Progression be given, assume the
numbers as \(\frac{a}{r}\), a and ar. Here common ratio is r.

(ii) If the product of four numbers in Geometric Progression be given, assume the numbers as \(\frac{a}{r^{3}}\), \(\frac{a}{r}\), ar and ar\(^{3}\). Here common ratio is r\(^{2}\).

(iii) If the product of five numbers in Geometric Progression be given, assume the numbers as \(\frac{a}{r^{2}}\), \(\frac{a}{r}\), a, ar and ar\(^{2}\). Here common ratio is r.

(iv) If the product of the numbers is not given, then the numbers are taken as a, ar, ar\(^{2}\), ar\(^{3}\), ar\(^{4}\), ar\(^{5}\), .....................

**6.** A series of the form a + ar + ar\(^{2}\) + ar\(^{3}\)...... + ar\(^{n}\) + ............... ∞ is
called an infinite geometric series.

**7.** The sum of an infinite Geometric Progression whose first term 'a' and
common ratio 'r' (-1 < r < 1 i.e., |r| < 1) is

S = \(\frac{a}{1 - r}\)

i.e., If -1 < r < 1, then a + ar + ar\(^{2}\) + ar\(^{3}\) + ar\(^{4}\) + .............. ∞ = \(\frac{a}{1 - r}\)

**●** **Geometric Progression**

**Definition of****Geometric Progression****General Form and General Term of a Geometric Progression****Sum of n terms of a Geometric Progression****Definition of Geometric Mean****Position of a term in a Geometric Progression****Selection of Terms in Geometric Progression****Sum of an infinite Geometric Progression****Geometric Progression Formulae****Properties of Geometric Progression****Relation between Arithmetic Means and Geometric Means****Problems on Geometric Progression**

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