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Difficulty: MediumArithmetic and Geometric Sequences and Series

A geometric sequence consists of positive terms. The first term of the sequence is 33, and the sum of the first 33 terms is 3939. What is the value of the 44 th term of this sequence?

  1. A
    33
  2. 81Answer
  3. C
    120
  4. D
    192
  5. E
    243

Answer

81
The sum of the first three terms of a geometric sequence is given by S3=a1(1+r+r2)S_3 = a_1(1 + r + r^2). Substituting the first term a1=3a_1 = 3 and the sum S3=39S_3 = 39 gives 3(1+r+r2)=393(1 + r + r^2) = 39, which simplifies to r2+r−12=0r^2 + r - 12 = 0. Factoring this quadratic equation yields (r−3)(r+4)=0(r - 3)(r + 4) = 0. Since the sequence consists of positive terms, the common ratio must be positive, which gives r=3r = 3. The fourth term of the sequence is then calculated using the formula a4=a1r3=3⋅33=81a_4 = a_1 r^3 = 3 \cdot 3^3 = 81.

Step-by-Step Solution

1
Write the expression for the sum of the first 3 terms of a geometric sequence and set it equal to the given sum.
S3=a1(1+r+r2)=3(1+r+r2)=39S_3 = a_1(1 + r + r^2) = 3(1 + r + r^2) = 39
This sets up the equation needed to solve for the common ratio of the sequence.
2
Divide both sides of the equation by 3 and solve the resulting quadratic equation for the common ratio rr.
1+r+r2=13  ⟹  r2+r−12=0  ⟹  (r−3)(r+4)=0  ⟹  r=31 + r + r^2 = 13 \implies r^2 + r - 12 = 0 \implies (r - 3)(r + 4) = 0 \implies r = 3 (since terms must be positive)
This determines the common ratio of the geometric sequence.
3
Use the formula for the nn-th term of a geometric sequence, an=a1rn−1a_n = a_1 r^{n-1}, to find the value of the 4th term.
a4=3⋅33=3⋅27=81a_4 = 3 \cdot 3^3 = 3 \cdot 27 = 81
This computes the final value requested by the question.

Key Concept

Geometric sequence formulas for the sum of a finite number of terms and the n-th term
Estimated Time:1m 30s
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Topics

Arithmetic and Geometric Sequences and Series

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