The average of a series of five numbers, in which each term is greater than its previous term by 3, is 12. What is the value of the greatest term?

Options

A. 24
B. 12
C. 20
D. 18

Answer

To solve the problem step by step, we can follow these instructions:

1. **Understand the Problem**: We need to find the greatest term in a series of five numbers where each term increases by 3 from the previous term, and the average of these numbers is 12.

2. **Calculate the Sum of the Numbers**:
- The average of the five numbers is given as 12.
- The formula for the average is:
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
- Rearranging this gives us:
\[ \text{Sum of observations} = \text{Average} \times \text{Number of observations} \]
- Substituting the values:
\[ \text{Sum} = 12 \times 5 = 60 \]

3. **Define the Numbers**:
- Let the first number be \( x \).
- The series of numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 3 \)
- Third number: \( x + 6 \)
- Fourth number: \( x + 9 \)
- Fifth number: \( x + 12 \)

4. **Set Up the Equation for the Sum**:
- The sum of these five numbers can be written as:
\[ x + (x + 3) + (x + 6) + (x + 9) + (x + 12) \]
- Simplifying this gives:
\[ 5x + (3 + 6 + 9 + 12) = 5x + 30 \]

5. **Equate the Sum to 60**:
- We know from step 2 that the sum is 60, so we set up the equation:
\[ 5x + 30 = 60 \]

6. **Solve for \( x \)**:
- Subtract 30 from both sides:
\[ 5x = 60 - 30 \]
\[ 5x = 30 \]
- Divide both sides by 5:
\[ x = 6 \]

7. **Find the Greatest Term**:
- The greatest term in the series is the fifth number:
\[ x + 12 = 6 + 12 = 18 \]

8. **Conclusion**:
- The value of the greatest term is **18**.