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Difficulty: MediumArithmetic and Geometric Progressions (AP and GP)

The sum of the first three terms of an increasing geometric progression of positive real numbers is 2121, and the sum of their squares is 189189. What is the common ratio of this progression?

Answer: 2

Answer

The common ratio of the geometric progression is 2.
Let the terms be aa, arar, and ar2ar^2. The given conditions yield a(1+r+r2)=21a(1+r+r^2) = 21 and a2(1+r2+r4)=189a^2(1+r^2+r^4) = 189. Squaring the first equation gives a2(1+r+r2)2=441a^2(1+r+r^2)^2 = 441. Dividing the sum of squares equation by this squared equation gives 1−r+r21+r+r2=189441=37\frac{1-r+r^2}{1+r+r^2} = \frac{189}{441} = \frac{3}{7}. Simplifying 7(1−r+r2)=3(1+r+r2)7(1-r+r^2) = 3(1+r+r^2) results in 2r2−5r+2=02r^2 - 5r + 2 = 0, which factors into (2r−1)(r−2)=0(2r-1)(r-2) = 0. Because the progression is increasing, r>1r > 1, making r=2r = 2 the correct common ratio.

Step-by-Step Solution

1
Formulate algebraic expressions for the sum of terms and sum of squares.
a(1+r+r2)=21a(1 + r + r^2) = 21 and a2(1+r2+r4)=189a^2(1 + r^2 + r^4) = 189
The first three terms of any geometric progression can be expressed as aa, arar, and ar2ar^2.
2
Eliminate the first term aa by squaring the first equation and dividing.
a2(1+r2+r4)a2(1+r+r2)2=189441  ⟹  (1+r+r2)(1−r+r2)(1+r+r2)2=37\frac{a^2(1 + r^2 + r^4)}{a^2(1 + r + r^2)^2} = \frac{189}{441} \implies \frac{(1 + r + r^2)(1 - r + r^2)}{(1 + r + r^2)^2} = \frac{3}{7}
Using the algebraic factorization 1+r2+r4=(1+r+r2)(1−r+r2)1 + r^2 + r^4 = (1 + r + r^2)(1 - r + r^2) allows cancellation of a2a^2 and (1+r+r2)(1 + r + r^2).
3
Solve the resulting equation for the common ratio rr.
7(1−r+r2)=3(1+r+r2)  ⟹  4r2−10r+4=0  ⟹  2r2−5r+2=07(1 - r + r^2) = 3(1 + r + r^2) \implies 4r^2 - 10r + 4 = 0 \implies 2r^2 - 5r + 2 = 0
Cross-multiplying reduces the ratio to a standard quadratic equation.
4
Factor the quadratic equation and select the correct root.
(2r−1)(r−2)=0  ⟹  r=2 or r=0.5(2r - 1)(r - 2) = 0 \implies r = 2 \text{ or } r = 0.5
Since the geometric progression is specified as increasing, the common ratio must be greater than 1 (r=2r = 2).

Key Concept

Geometric progression term representations, sum formulas, and algebraic identity factorization.

Alternative Method

Find aa and rr by testing factors of 2121: 21=3×721 = 3 \times 7, so the terms could be 3,6,123, 6, 12 (a=3,r=2a=3, r=2). Check squares: 32+62+122=9+36+144=1893^2 + 6^2 + 12^2 = 9 + 36 + 144 = 189, which confirms r=2r = 2.
Estimated Time:1m 30s
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Topics

Arithmetic and Geometric Progressions (AP and GP)

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