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
11 and 12 Grade Math
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