Exploring Geometric Progressions: Three Numbers Whose Sum and Product Are Given

In mathematics, geometric progressions (GP) are sequences 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. This article delves into a specific problem where the sum and product of three numbers in a geometric progression are given, illustrating how to solve such problems using algebraic methods.

The Problem at Hand

The problem posed is to find three numbers in geometric progression whose sum is (10.5) and whose product is (27). Let's denote the three numbers as (a/r, a, ar) where (a) is the first term and (r) is the common ratio.

Understanding the Given Information

First, let's examine the given information:

The sum of the numbers is (10.5). The product of the numbers is (27).

Solving the Sum Equation

The sum of the three numbers in geometric progression can be expressed as:

[frac{a}{r} a ar 10.5]

Factoring out (a), we get:

[a left(frac{1}{r} 1 rright) 10.5]

Solving the Product Equation

The product of the three numbers is given by:

[frac{a}{r} cdot a cdot ar a^3 27]

Since (a^3 27), we can solve for (a):

[a 3]

Substituting (a 3) into the Sum Equation

Substituting (a 3) into the sum equation, we get:

[3 left(frac{1}{r} 1 rright) 10.5]

Dividing both sides by 3, we have:

[frac{1}{r} 1 r 3.5]

Rearranging the equation, we get a quadratic equation:

[1 r frac{1}{r} 3.5]

Multiplying through by (r) to clear the fraction, we get:

[r r^2 1 3.5r]

Bringing all terms to one side, we obtain:

[r^2 - 2.5r 1 0]

This is a standard quadratic equation. Using the quadratic formula (r frac{-b pm sqrt{b^2 - 4ac}}{2a}) where (a 1), (b -2.5), and (c 1), we find the values of (r):

[r frac{2.5 pm sqrt{6.25 - 4}}{2}] [r frac{2.5 pm sqrt{2.25}}{2}] [r frac{2.5 pm 1.5}{2}]

This gives us two possible values for (r):

[r 2 quad text{or} quad r 0.5]

Finding the Values of the Numbers

Substituting these values of (r) back into the original sequence formula, we get:

If (r 2), the numbers are ( frac{3}{2}, 3, 6 ). If (r 0.5), the numbers are (6, 3, frac{3}{2} ).

Both sequences are correct as they represent the same set of numbers in reverse order.

Conclusion

In conclusion, the three numbers in geometric progression that satisfy the conditions of summing to (10.5) and having a product of (27) are (1.5, 3, 6) (and their reverse order (6, 3, 1.5)). This problem highlights the application of algebra and the properties of geometric sequences in solving complex mathematical problems.

Related Keywords

geometric progression sum and product sequences

Understanding these concepts can be useful in various fields such as finance, physics, and engineering. If you're looking to deepen your knowledge in these areas, further exploration of geometric sequences and series can provide valuable insights.

References:

Geometric Progression Arithmetic and Geometric Sequences