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 article will illustrate a method to find three numbers in geometric progression when given their sum and product.

Problem

The problem presented is: find three numbers in GP such that their sum is 30 and their product is 216.

Solution

Let's denote the three numbers in GP as ( frac{a}{r} ), ( a ), and ( ar ). Here, ( r ) is the common ratio.

Step 1: Setting Up the Equations

1. The sum of the numbers: [ frac{a}{r} a ar 30 ] 2. The product of the numbers: [ left(frac{a}{r}right) cdot a cdot (ar) 216 ]

Simplifying the product equation:

[ a^3 r^3 216 ] [ a^3 216 ] [ a 6 ]

Substituting ( a 6 ) into the sum equation:

[ frac{6}{r} 6 6r 30 ] Multiplying the entire equation by ( r ) to clear the fraction:

[ 6 6r 6r^2 30r ] Rearranging terms:

[ 6r^2 - 24r 6 0 ]

Dividing by 6:

[ r^2 - 4r 1 0 ]

Step 2: Solving the Quadratic Equation

Using the quadratic formula ( r frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a 1 ), ( b -4 ), and ( c 1 ):

[ r frac{4 pm sqrt{16 - 4}}{2} ] [ r frac{4 pm sqrt{12}}{2} ] [ r frac{4 pm 2sqrt{3}}{2} ] [ r 2 pm sqrt{3} ]

Step 3: Finding the Values of ( a )

Using ( r 2 sqrt{3} ) and ( r 2 - sqrt{3} ):

[ a frac{6}{2 sqrt{3}} cdot frac{2 - sqrt{3}}{2 - sqrt{3}} frac{6(2 - sqrt{3})}{4 - 3} 6(2 - sqrt{3}) ] [ a frac{6}{2 - sqrt{3}} cdot frac{2 sqrt{3}}{2 sqrt{3}} frac{6(2 sqrt{3})}{4 - 3} 6(2 sqrt{3}) ]

Step 4: Finding the Three Numbers

For ( r 2 sqrt{3} ) and ( a 6(2 - sqrt{3}) ):

[ frac{a}{r} frac{6(2 - sqrt{3})}{2 sqrt{3}} cdot frac{2 - sqrt{3}}{2 - sqrt{3}} frac{6(2 - sqrt{3})^2}{4 - 3} 6(4 - 4sqrt{3} 3) 6(7 - 4sqrt{3}) approx 3.608 ] [ ar 6(2 - sqrt{3})(2 sqrt{3}) 6(4 - 3) 6 cdot 1 6 ] [ ar^2 6(2 - sqrt{3})^2 cdot (2 sqrt{3}) 6(7 - 4sqrt{3})(2 sqrt{3}) 6(14 7sqrt{3} - 8sqrt{3} - 12) 6(2 - sqrt{3}) 16.392 ]

For ( r 2 - sqrt{3} ) and ( a 6(2 sqrt{3}) ):

[ frac{a}{r} frac{6(2 sqrt{3})}{2 - sqrt{3}} cdot frac{2 sqrt{3}}{2 sqrt{3}} frac{6(2 sqrt{3})^2}{4 - 3} 6(4 4sqrt{3} 3) 6(7 4sqrt{3}) approx 16.392 ] [ ar 6(2 sqrt{3})(2 - sqrt{3}) 6(4 - 3) 6 cdot 1 6 ] [ ar^2 6(2 sqrt{3})^2 cdot (2 - sqrt{3}) 6(7 4sqrt{3})(2 - sqrt{3}) 6(14 - 7sqrt{3} 8sqrt{3} - 12) 6(2 sqrt{3}) 3.608 ]

Final Result

The three numbers are approximately:

Case 1: ( 3.608, 6, 16.392 )

Case 2: ( 16.392, 6, 3.608 )

Both cases yield the same set of numbers, confirming the solution is correct.