The value of `sqrt(97 xx 98 xx 99 xx100+1)` is equal to

Options

A. 9901
B. 9891
C. 9801
D. 9701

Answer

To solve the problem of finding the value of \( \sqrt{97 \times 98 \times 99 \times 100 + 1} \), we can follow these steps:

### Step 1: Recognize the expression
We start with the expression:
\[ \sqrt{97 \times 98 \times 99 \times 100 + 1} \]

### Step 2: Rewrite the product
Notice that \( 97 \times 98 \times 99 \times 100 \) can be rewritten in terms of a variable. Let \( x = 98.5 \) (the average of the four numbers). Then we can express the numbers as:
- \( 97 = x - 1.5 \)
- \( 98 = x - 0.5 \)
- \( 99 = x + 0.5 \)
- \( 100 = x + 1.5 \)

### Step 3: Use the identity
We can use the identity:
\[ (a-1)(a)(a+1)(a+2) = (a^2 - 1)(a^2 + 2a) = a^4 + a^2 - 2 \]
In our case, we can set \( a = 98 \):
\[ (97)(98)(99)(100) = (98^2 - 1)(98^2 + 2 \times 98) = (9604 - 1)(9604 + 196) = 9603 \times 9800 \]

### Step 4: Simplify the expression
Now, we can express the original expression:
\[ 97 \times 98 \times 99 \times 100 + 1 = (97 \times 98 \times 99 \times 100) + 1 \]
This can be rewritten as:
\[ (98^2 - 1)(98^2 + 2 \times 98) + 1 \]

### Step 5: Calculate the square root
Now, we can express the entire expression as a perfect square:
\[ \sqrt{(97 \times 98 \times 99 \times 100) + 1} = \sqrt{(98 \times 99)^2} = 98 \times 99 + 1 = 9701 \]

### Final Answer
Thus, the value of \( \sqrt{97 \times 98 \times 99 \times 100 + 1} \) is:
\[ \boxed{9701} \]