https://file.aiquickdraw.com/mathbot/file/1723870740044yk15pbpm.jpg

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

To find the second term in a geometric sequence given the first term \( f(1) = 3 \) and the third term \( f(3) = 75 \), 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 \( r \).

The general formula for the \( n \)th term is:
\[
f(n) = f(1) \times r^{n-1}
\]

### Step 2: Apply the formula to the given terms
- For the first term (\( n = 1 \)):
  \[
  f(1) = 3
  \]
- For the third term (\( n = 3 \)):
  \[
  f(3) = f(1) \times r^{3-1} = 3 \times r^2 = 75
  \]

### Step 3: Solve for the common ratio \( r \)
\[
3 \times r^2 = 75
\]
Divide both sides by 3:
\[
r^2 = 25
\]
Take the square root of both sides:
\[
r = 5
\]

### Step 4: Find the second term
Using the common ratio \( r = 5 \):
\[
f(2) = f(1) \times r^{2-1} = 3 \times 5 = 15
\]

**Answer:** The second term \( f(2) \) 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 \( n \) terms of a geometric sequence?
3. How can you find the \( n \)th term if you know the sum of the first \( n \) 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 \( n \) increases if the common ratio \( r \) 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.

Learn how to find the second term in a geometric sequence given the first and third terms. Follow a step-by-step solution involving the formula for the nth term of a geometric sequence. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation