What is the nth Term of a Finite GP?

The nth term of a finite Geometric Progression (GP) is given by the formula: T_n = ar^n-1, where 'a' is the first term, and 'r' is the common ratio.

Understanding the Formula

A Geometric Progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor, called the common ratio (r). Understanding the components of the formula is key to calculating any term in a GP.

• T_n: Represents the nth term of the GP. This is the value you are trying to find.
• a: Represents the first term of the GP, which is the starting point of the sequence.
• r: Represents the common ratio of the GP. This is the constant factor by which each term is multiplied to get the next term. It can be calculated by dividing any term by its preceding term (r = T_n/T_n-1).
• n: Represents the position of the term you want to find in the sequence (e.g., if you want to find the 5th term, n = 5).

How to Use the Formula

To find the nth term of a GP, you need to know the first term (a), the common ratio (r), and the term number (n). Then, simply substitute these values into the formula T_n = ar^n-1 and calculate the result.

Example

Let's say we have a GP with the first term a = 2 and a common ratio r = 3. We want to find the 4th term (T₄).

Using the formula:

T₄ = ar^n-1
T₄ = 2 3^4-1
T₄ = 2 3³
T₄ = 2 * 27
T₄ = 54

Therefore, the 4th term of this GP is 54.

Finding the nth term from the end of a finite GP

If 'm' is the total number of terms in a finite GP, the nth term from the end is given by:

ar^m-n

Here, 'a' is the first term, 'r' is the common ratio, 'm' is the total number of terms, and 'n' is the position from the end you want to find.

For example, consider the GP: 2, 4, 8, 16, 32 (where a=2, r=2, m=5). Find the 2nd term from the end.

Using the formula ar^m-n:

2 2^5-2 = 2 2³ = 2 * 8 = 16.

The 2nd term from the end is 16.