Derive the formula for calculation of geometric progression.
The formula for the general term (n-th term) of a geometric progression is given by:
an = a * r^(n-1),
where:
an = the n-th term,
a = the first term,
r = the common ratio, and
n = the position of the term in the sequence.
To derive this formula, we can start by considering the ratio between consecutive terms, which is denoted by r. Let's say we have the first term a, and the second term is ar. Since the second term is obtained by multiplying the first term by the common ratio r, we can express it as ar.
To find the third term, we multiply the second term by the common ratio r again, giving arr. Continuing this process, we can observe that the n-th term is obtained by multiplying the first term a by the common ratio r a total of (n-1) times.
So, by multiplying a by r^(n-1), we obtain the formula for the general term of a geometric progression:
an = a * r^(n-1).
To derive the formula for the calculation of a geometric progression, let's start by understanding the definition of a geometric progression.
A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant factor called the common ratio (r).
Let's denote the first term as 'a', and the common ratio as 'r'. The terms of the geometric progression can be written as follows:
a, ar, ar^2, ar^3, ...
To find the nth term of a geometric progression, we can use the following formula:
T_n = a * r^(n-1)
where:
- T_n is the nth term of the geometric progression
- a is the first term
- r is the common ratio
- n is the position of the term in the sequence
This formula works because each term is obtained by multiplying the previous term by the common ratio 'r'. So, to find the nth term, we multiply the first term 'a' by the common ratio 'r' (n-1) times, as there are (n-1) multiplications needed to obtain the nth term.
This formula can also be rearranged to find the common ratio (r), given the first term (a) and the nth term (T_n). Rearranging the formula, we get:
r = (T_n / a)^(1/(n-1))
So, the formula for calculating the nth term of a geometric progression is T_n = a * r^(n-1), while the formula for finding the common ratio (r) is r = (T_n / a)^(1/(n-1)).