The average of 3 consecutive natural numbers(which are in increasing order)is k.if two more consecutive number,just next first set of numbers is added then the new average becomes:

Options

A. k+2
B. k+1
C. 2k+1//2
D. 2k-1

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the three consecutive natural numbers
Let the three consecutive natural numbers be:
- \( n \)
- \( n + 1 \)
- \( n + 2 \)

### Step 2: Calculate the average of these three numbers
The average \( k \) of these three numbers can be calculated as follows:
\[ k = \frac{n + (n + 1) + (n + 2)}{3} \]
Simplifying this:
\[ k = \frac{3n + 3}{3} = n + 1 \]

### Step 3: Identify the next two consecutive natural numbers
The next two consecutive natural numbers after \( n + 2 \) are:
- \( n + 3 \)
- \( n + 4 \)

### Step 4: Calculate the new average when these two numbers are added
Now, we have five numbers in total:
- \( n \)
- \( n + 1 \)
- \( n + 2 \)
- \( n + 3 \)
- \( n + 4 \)

The new average can be calculated as:
\[ \text{New Average} = \frac{n + (n + 1) + (n + 2) + (n + 3) + (n + 4)}{5} \]
Simplifying this:
\[ \text{New Average} = \frac{5n + 10}{5} = n + 2 \]

### Step 5: Relate the new average to \( k \)
Since we found that \( k = n + 1 \), we can express the new average in terms of \( k \):
\[ \text{New Average} = n + 2 = (n + 1) + 1 = k + 1 \]

### Conclusion
Thus, the new average becomes \( k + 1 \).

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