Introduction to Geometric Progression

A geometric progression (GP), 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 fixed, non-zero number called the common ratio. The first term of the sequence is often referred to as a, and the nth term can be calculated using the formula: [ T_n ar^{n-1} ]

The sum of the first n terms of a GP is given by: [ S_n frac{a(r^n - 1)}{r - 1} quad text{if } r eq 1 ]

Determining the First Term from Given Conditions

Let's explore a problem where specific terms of a GP are given, and we need to determine the first term.

Problem Statement:

We are given that the sum of the first eight terms of a GP is 6560 and the common ratio is 3. We need to find the first term.

Solution:

Given: [ S_8 6560 quad text{and} quad r 3 ]

We use the formula for the sum of the first n terms of a GP: [ S_n frac{a(r^n - 1)}{r - 1} ]

Substitute the given values: [ 6560 frac{a(3^8 - 1)}{3 - 1} ] Simplify the equation: [ 6560 frac{a(6561 - 1)}{2} ]

[ 6560 frac{a times 6560}{2} ]

[ 13120 a times 6560 ]

[ a frac{13120}{6560} ] [ a 2 ]

Therefore, the first term of the GP is 2.

Additional Perspectives on Finding the First Term

Let's consider another approach using the terms provided.

Second Method:

Given the third term and the seventh term of the GP and their sum is 3267, and the common ratio is 3, we can proceed as follows:

The third term is: [ ar^2 a times 3^2 ] The seventh term is: [ ar^6 a times 3^6 ] We know these terms sum up to 3267: [ ar^2 ar^6 3267 ] Substitute the values: [ a times 9 a times 729 3267 ]

[ 9a 729a 3267 ]

[ 738a 3267 ]

[ a frac{3267}{738} ] [ a frac{363}{82} approx 4.426 ]

Thus, the first term is approximately 4.426.

Conclusion

We have explored two methods to find the first term of a geometric progression given specific conditions. Both methods lead to the first term being 2 (when the sum of the first eight terms is 6560) and approximately 4.426 (when the sum of the third and seventh terms is 3267).

Understanding these methods is crucial for other problems involving geometric progressions, such as finding specific terms, sum of terms, or common ratio.