If average of 29 consecutive even numbers is 446. then find difference between first and last number ?

Options

A. 58
B. 56
C. 60
D. 62

Answer

To find the difference between the first and last of 29 consecutive even numbers with an average of 446, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Average of Consecutive Even Numbers**:
The average of a set of numbers is the sum of the numbers divided by the count of the numbers. Here, we have 29 consecutive even numbers, and their average is given as 446.

2. **Calculate the Total Sum of the Numbers**:
Since the average is 446 and there are 29 numbers, we can find the total sum (S) of these numbers:
\[ S = \text{Average} \times \text{Number of terms} = 446 \times 29 \]
\[ S = 12934 \]

3. **Define the First Even Number**:
Let the first even number be \( A \). The sequence of the 29 consecutive even numbers can be expressed as:
\[ A, A+2, A+4, \ldots, A+56 \]
(since the difference between each consecutive even number is 2).

4. **Identify the Last Even Number**:
The last (29th) even number in this sequence can be expressed as:
\[ A_{29} = A + (29 - 1) \times 2 = A + 56 \]

5. **Calculate the Difference Between the First and Last Number**:
The difference between the last and the first number is:
\[ A_{29} - A = (A + 56) - A = 56 \]

6. **Conclusion**:
Therefore, the difference between the first and last number is 56.

### Final Answer:
The difference between the first and last number is **56**.

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