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Chapter 22: Problem 697

Find three numbers in geometric progression whose sum is 19, and whose product is 216 .

Short Answer

Expert verified
The three numbers in geometric progression are 3, \( \frac{3}{2} \), and \( \frac{3}{4} \).

Step by step solution

01

Rewrite equations using the variable, a

Rewrite the product equation in terms of "a", by factoring a: a * ar * ar^2 = 216 鈬� a^3 * r^3 = 216
02

Solve one equation for one variable

Solve the sum equation for "r": a + ar + ar^2 = 19 鈬� r = (19 - a - ar^2) / a
03

Substitute found expression in the other equation

Substitute the expression found in Step 2 into the rewritten product equation: a^3 * ((19 - a - ar^2) / a)^3 = 216
04

Simplify the equation

Simplify the equation resulting from the substitution: (19 - a - ar^2)^3 = 216a^2
05

Observe that 216 is a perfect cube

Recognize that 216 is a perfect cube (6^3) and try with integer values of a.
06

Test integer values

Test integer values of a: 1 to 6 When a=3: (19 - 3 - 3r^2)^3 = 216(3)^2 (16 - 3r^2)^3 = 1944 Let's test if the left-hand side can produce an equation equal to 1944. If r=1, the left-hand side would be (13^3) which is too high. Let's try r<1, for example r=1/2. (16 - 3(1/2)^2)^3 = 1944 (14)^3 = 1944 We have successfully found values for a and r that satisfy both the sum and the product equation: a = 3 and r = 1/2.
07

Find the three numbers in geometric progression

Use the values of a and r to find the three numbers: 1. First number: a = 3 2. Second number: ar = 3 * (1/2) = 3/2 3. Third number: ar^2 = 3 * (1/2)^2 = 3/4 The three numbers in geometric progression are 3, \( \frac{3}{2} \), and \( \frac{3}{4} \).

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Most popular questions from this chapter

The sum of an infinite number of terms in geometric progression is 15 , and the sum of their squares is 45 ; find the sequence. Assume that the common ratio of the G.P. is less than 1 .

The sums of n terms of two arithmetic progressions are in the ratio of \(7 \mathrm{n}+1: 4 \mathrm{n}+27 ;\) find the ratio of their 11 th terms

Insert three harmonic means between \(1 / 10\) and \(1 / 42\).

Find the sum of the first 20 terms of the arithmetic progression \(-9,-3,3, \ldots\)

If \(\mathrm{x}<1\), sum the series \(1+2 \mathrm{x}+3 \mathrm{x}^{2}+4 \mathrm{x}^{3}+\ldots\) to infinity.
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