Math Problem Statement

The first and third terms of a geometric sequence are shown in the table below. What is the second term?

Solution

To find the second term in a geometric sequence given the first term f(1)=3f(1) = 3 and the third term f(3)=75f(3) = 75, follow these steps:

Step 1: Set up the formula for a geometric sequence

In a geometric sequence, each term is found by multiplying the previous term by a common ratio rr.

The general formula for the nnth term is: f(n)=f(1)×rn1f(n) = f(1) \times r^{n-1}

Step 2: Apply the formula to the given terms

  • For the first term (n=1n = 1): f(1)=3f(1) = 3
  • For the third term (n=3n = 3): f(3)=f(1)×r31=3×r2=75f(3) = f(1) \times r^{3-1} = 3 \times r^2 = 75

Step 3: Solve for the common ratio rr

3×r2=753 \times r^2 = 75 Divide both sides by 3: r2=25r^2 = 25 Take the square root of both sides: r=5r = 5

Step 4: Find the second term

Using the common ratio r=5r = 5: f(2)=f(1)×r21=3×5=15f(2) = f(1) \times r^{2-1} = 3 \times 5 = 15

Answer: The second term f(2)f(2) is 15.

Would you like to explore more about geometric sequences, or do you have other questions?

Related Questions:

  1. How would the sequence change if the common ratio were negative?
  2. What is the formula for the sum of the first nn terms of a geometric sequence?
  3. How can you find the nnth term if you know the sum of the first nn terms?
  4. Can a geometric sequence have a common ratio of 1? What would it look like?
  5. What is the product of the first and third terms in this sequence?
  6. How do geometric sequences differ from arithmetic sequences?
  7. What happens to the terms of a geometric sequence as nn increases if the common ratio rr is between 0 and 1?
  8. What would be the fourth term in this sequence?

Tip: In geometric sequences, recognizing the relationship between the terms through the common ratio helps solve for unknown terms quickly.

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Math Problem Analysis

Mathematical Concepts

Geometric Sequences

Formulas

Formula for the nth term of a geometric sequence

Theorems

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Suitable Grade Level

Grades 9-12