Understanding Geometric Progression (GP)

Geometric progression, or GP, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). This can be represented mathematically as Tn T1r^(n-1).

Example: Given First Term is 6 and Common Ratio is 8

Let's explore a geometric progression where the first term (a1) is 6 and the common ratio (r) is 8.

Finding the Third Term (a3)

The formula for the nth term of a GP is Tn a1r^(n-1). For the third term, n 3.

T3 a1r^(3-1) 6 times; 8^2 6 times; 64 384

Sequence Visualization

Term (n) Value 1 6 2 48 3 384 4 3072 5 24576

The Formula and Its Application

The general formula for the nth term of a GP is important for solving such problems:

Tn a1r^(n-1)

To find the fifth term (T5), we substitute n 5 into the formula:

T5 6 times; 8^4 6 times; 4096 24576

Geometric Progression and Its Multiplicative Nature

Each term in a GP is a product of the previous term and the common ratio. Therefore, starting from the first term, the sequence can be generated as follows:

a1 6 a2 6 times; 8 48 a3 48 times; 8 384 a4 384 times; 8 3072 a5 3072 times; 8 24576

Conclusion

By understanding the concept of geometric progression and applying the appropriate formula, we can easily determine the value of any term in the sequence, as demonstrated in the example above. The common ratio determines the multiplicative relationship between consecutive terms, making it a fundamental aspect of geometric sequences.