Introduction to Geometric Progression

Geometric progression (GP) 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. This common ratio is denoted by r. The n-th term of a GP can be expressed using the formula:

an a rn-1

Problem Statement

The n-th term of a geometric progression is given by:

an a rn-1

We are given a geometric progression with the common ratio r sqrt{3} and the sixth term a_6 27. Our task is to determine the first term a.

Step-by-Step Solution

1. **Express the Sixth Term Using the Formula:**

a_6 a r^{6-1} a r^5

2. **Substitute the Known Values:**

27 a (sqrt{3})^5

3. **Simplify the Expression: **

Calculate (sqrt{3})^5 which is equal to 3^{5/2} or 3^2 sqrt{3} 9sqrt{3}.

27 a 9sqrt{3}

4. **Solve for the First Term a:**

a frac{27}{9sqrt{3}} frac{3}{sqrt{3}} sqrt{3}

Therefore, the first term of the geometric progression is:

boxed{sqrt{3}}

Alternative Approaches

Another way to solve this problem involves understanding that going from the 6th term to the 1st term, the ratio changes to 1/sqrt{3}. Hence:

a_6 frac{1}{sqrt{3}}^5 27 frac{1}{9sqrt{3}} 3/sqrt{3} sqrt{3}

This confirms the first term is sqrt{3}.

Conclusion

This step-by-step solution simplifies finding the first term of a geometric progression when given the sixth term and the common ratio. Understanding these basic principles can be applied to solve similar problems effectively.

Related Topics

For further exploration, consider studying more topics in:

Geometric Progressions and Their Formulas Sequences and Series in Mathematics Common Ratio in Geometric Progression