Geometric progression, also known as a geometric sequence, is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In simpler terms, it is a sequence where each term is obtained by multiplying the previous term by a constant.
Geometric progression involves several key concepts:
To understand geometric progression, let's consider an example sequence: 2, 4, 8, 16, 32, ...
In this sequence, the common ratio is 2, the first term is 2, and the nth term can be calculated using the formula: a_n = a_1 * r^(n-1), where a_n represents the nth term, a_1 is the first term, r is the common ratio, and n is the position of the term.
The formula for finding the nth term of a geometric progression is given by:
a_n = a_1 * r^(n-1)
Here, a_n represents the nth term, a_1 is the first term, r is the common ratio, and n is the position of the term.
To apply the geometric progression formula, you need to know the values of the first term (a_1), the common ratio (r), and the position of the term (n). Simply substitute these values into the formula:
a_n = a_1 * r^(n-1)
By plugging in the known values, you can calculate the desired term in the sequence.
There is no specific symbol exclusively used for geometric progression. However, the terms in a geometric progression are often denoted as a_1, a_2, a_3, ..., a_n, where a_1 represents the first term and a_n represents the nth term.
There are several methods to work with geometric progressions:
Example 1: Consider a geometric progression with a first term (a_1) of 3 and a common ratio (r) of 2. Find the 5th term (a_5).
Solution: Using the formula a_n = a_1 * r^(n-1), we can substitute the given values: a_5 = 3 * 2^(5-1) = 3 * 2^4 = 3 * 16 = 48
Example 2: Find the sum of the first 6 terms of a geometric progression with a first term of 2 and a common ratio of 3.
Solution: Using the formula S_n = a_1 * (1 - r^n) / (1 - r), we can substitute the given values: S_6 = 2 * (1 - 3^6) / (1 - 3) = 2 * (1 - 729) / (-2) = 2 * (-728) / (-2) = 728
Q: What happens if the common ratio in a geometric progression is 1? A: If the common ratio is 1, all terms in the sequence will be equal to the first term, resulting in a constant sequence.
Q: Can the common ratio in a geometric progression be negative? A: Yes, the common ratio can be negative. In such cases, the sequence alternates between positive and negative terms.
Q: What is the sum of an infinite geometric progression? A: The sum of an infinite geometric progression exists only if the absolute value of the common ratio is less than 1. In that case, the sum is given by S = a_1 / (1 - r).
Q: Can geometric progressions have fractions or decimals as terms? A: Yes, geometric progressions can have fractional or decimal terms, depending on the values of the first term and the common ratio.