If `a = sqrt((2013)^(2) + 2013 + 2014)`, then the value of a is

Options

A. 1002
B. 1007
C. 2013
D. 2014

Answer

To solve the expression \( a = \sqrt{(2013)^2 + 2013 + 2014} \), we will follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ a = \sqrt{(2013)^2 + 2013 + 2014} \]

### Step 2: Simplify inside the square root
We can combine the terms inside the square root:
\[ (2013)^2 + 2013 + 2014 = (2013)^2 + 2013 + 2014 \]

### Step 3: Combine like terms
Notice that we can rewrite \( 2014 \) as \( 2013 + 1 \):
\[ (2013)^2 + 2013 + (2013 + 1) = (2013)^2 + 2013 + 2013 + 1 = (2013)^2 + 2 \times 2013 + 1 \]

### Step 4: Recognize the perfect square
The expression \( (2013)^2 + 2 \times 2013 + 1 \) can be factored as:
\[ (2013 + 1)^2 \]
Thus, we have:
\[ a = \sqrt{(2013 + 1)^2} \]

### Step 5: Simplify the square root
Taking the square root of a square gives us:
\[ a = 2013 + 1 = 2014 \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{2014} \]

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