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Difficulty: MediumArithmetic and Geometric Sequences and Series

The sum of the first three terms of an increasing geometric sequence of positive numbers is 2121, and the sum of the squares of these same three terms is 189189. What is the first term of the sequence?

  1. A
    2
  2. 3Answer
  3. C
    6
  4. D
    9
  5. E
    12

Answer

The first term of the sequence is 3.
By setting up the equations a(1+r+r2)=21a(1+r+r^2) = 21 and a2(1+r2+r4)=189a^2(1+r^2+r^4) = 189, we use the identity 1+r2+r4=(1+r+r2)(1−r+r2)1+r^2+r^4 = (1+r+r^2)(1-r+r^2) to find a(1−r+r2)=9a(1-r+r^2) = 9. Subtracting this from the first equation gives 2ar=122ar = 12, so ar=6ar = 6. Solving for rr gives r=2r = 2 for an increasing sequence, which yields a=3a = 3.

Step-by-Step Solution

1
Set up the algebraic equations for the sum of terms and sum of squared terms.
Let the first three 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.
Standard representation of a geometric sequence with first term aa and common ratio rr.
2
Factor the sum of squares expression using algebraic identities.
Note that 1+r2+r4=(1+r+r2)(1−r+r2)1 + r^2 + r^4 = (1 + r + r^2)(1 - r + r^2). Thus, a2(1+r+r2)(1−r+r2)=189a^2(1 + r + r^2)(1 - r + r^2) = 189.
Factoring allows substitution of the first equation into the second.
3
Substitute a(1+r+r2)=21a(1 + r + r^2) = 21 into the factored equation.
Substituting 2121 gives 21⋅a(1−r+r2)=189  ⟹  a(1−r+r2)=921 \cdot a(1 - r + r^2) = 189 \implies a(1 - r + r^2) = 9.
Simplifies the second-degree term expression to a system of two linear equations in terms of aa and arar.
4
Subtract the simplified equation from the initial sum equation to isolate arar.
a(1+r+r2)−a(1−r+r2)=21−9  ⟹  2ar=12  ⟹  ar=6a(1 + r + r^2) - a(1 - r + r^2) = 21 - 9 \implies 2ar = 12 \implies ar = 6.
Eliminating terms isolates the product of the first term and ratio, which is the second term.
5
Solve for rr and find aa.
Substitute a=6ra = \frac{6}{r} into a(1+r+r2)=21a(1 + r + r^2) = 21 to get 6r2−15r+6=06r^2 - 15r + 6 = 0, which factors as (2r−1)(r−2)=0(2r - 1)(r - 2) = 0. Since the sequence is increasing, r=2r = 2, yielding a=3a = 3.
Determines the specific parameters satisfying the increasing constraint.

Key Concept

Properties and algebraic manipulation of terms in geometric sequences and series.
Estimated Time:2m 0s
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Topics

Arithmetic and Geometric Sequences and Series

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