Geometric Progression Formulae

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

11 and 12 Grade Math 

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