Math Problem Statement

Determine the common ratio and find the sum of the following geometric sequences: 1. 4, 12, 36, 108,... up to 7 terms. 2. 2, 6, 18, 54, 162,... up to 6 terms. 3. 7, 14, 28, 56,... up to 8 terms.

Solution

The image displays three geometric sequences, and the task is to find their common ratios and calculate the sum of the series for the given number of terms. Let's break down the sequences and solve them:

  1. Sequence 1: 4,12,36,108,4, 12, 36, 108, \ldots (up to 7 terms)

    • The first term a=4a = 4.
    • The common ratio r=124=3r = \frac{12}{4} = 3.
    • Formula for the sum of a geometric series: Sn=a(1rn)1r,r1S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1 For 7 terms: S7=4(137)13=4(12187)2=4(2186)2=4372S_7 = \frac{4(1 - 3^7)}{1 - 3} = \frac{4(1 - 2187)}{-2} = \frac{4(-2186)}{-2} = 4372 So, the sum of the first 7 terms is 4372.
  2. Sequence 2: 2,6,18,54,162,2, 6, 18, 54, 162, \ldots (up to 6 terms)

    • The first term a=2a = 2.
    • The common ratio r=62=3r = \frac{6}{2} = 3.
    • Using the same formula for 6 terms: S6=2(136)13=2(1729)2=2(728)2=728S_6 = \frac{2(1 - 3^6)}{1 - 3} = \frac{2(1 - 729)}{-2} = \frac{2(-728)}{-2} = 728 So, the sum of the first 6 terms is 728.
  3. Sequence 3: 7,14,28,56,7, 14, 28, 56, \ldots (up to 8 terms)

    • The first term a=7a = 7.
    • The common ratio r=147=2r = \frac{14}{7} = 2.
    • For 8 terms: S8=7(128)12=7(1256)1=7(255)1=1785S_8 = \frac{7(1 - 2^8)}{1 - 2} = \frac{7(1 - 256)}{-1} = \frac{7(-255)}{-1} = 1785 So, the sum of the first 8 terms is 1785.

Let me know if you need any more details or explanations!

Related Questions:

  1. How do you derive the formula for the sum of a geometric series?
  2. What happens when the common ratio is between 0 and 1?
  3. How would you find the sum if the series were infinite?
  4. Can a geometric series have a negative common ratio? How would it affect the sum?
  5. How do you calculate the sum of an arithmetic series?

Tip:

In geometric series, if the common ratio r<1|r| < 1, the series can converge to a finite value, even if there are infinitely many terms!

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Math Problem Analysis

Mathematical Concepts

Geometric Progression
Sum of a Geometric Series

Formulas

Sum of geometric series: S_n = a(1 - r^n) / (1 - r), where r ≠ 1

Theorems

Geometric Series Formula

Suitable Grade Level

Grades 9-12