Calculate the value of N in the given series and then find the value of x suing the given equation.
`99" "163" "N" "248" "273" "289`
If `sqrt(2N+17)=x`, then x equals

Options

A. 20.5
B. 20
C. 21.5
D. 21

Answer

To solve the problem step by step, we will first find the value of \( N \) in the given series and then use that value to find \( x \) using the equation provided.

### Step 1: Identify the Pattern in the Series
The given series is:
\[ 99, 163, N, 248, 273, 289 \]

We will calculate the differences between consecutive terms to find a pattern.

### Step 2: Calculate the Differences
1. \( 163 - 99 = 64 \)
2. \( N - 163 \) (we will calculate this later)
3. \( 248 - N \)
4. \( 273 - 248 = 25 \)
5. \( 289 - 273 = 16 \)

### Step 3: Analyze the Differences
From the differences we calculated:
- The difference between \( 99 \) and \( 163 \) is \( 64 \) (which is \( 8^2 \)).
- The difference between \( 273 \) and \( 248 \) is \( 25 \) (which is \( 5^2 \)).
- The difference between \( 289 \) and \( 273 \) is \( 16 \) (which is \( 4^2 \)).

### Step 4: Establish the Pattern
The differences appear to be squares of consecutive integers:
- \( 8^2 = 64 \)
- \( 7^2 = 49 \) (this should be the difference between \( 163 \) and \( N \))
- \( 6^2 = 36 \) (this should be the difference between \( N \) and \( 248 \))
- \( 5^2 = 25 \)
- \( 4^2 = 16 \)

### Step 5: Calculate \( N \)
Using the established pattern:
1. From \( 163 \) to \( N \):
\[ N = 163 + 49 = 212 \]
2. From \( N \) to \( 248 \):
\[ 248 = N + 36 \]
\[ 248 = 212 + 36 \] (this checks out)

Thus, the value of \( N \) is \( 212 \).

### Step 6: Find the Value of \( x \)
Now we will use the value of \( N \) to find \( x \) using the equation:
\[ \sqrt{2N + 17} = x \]

Substituting \( N = 212 \):
\[ \sqrt{2(212) + 17} = x \]
\[ \sqrt{424 + 17} = x \]
\[ \sqrt{441} = x \]
\[ x = 21 \]

### Final Answer
The value of \( x \) is \( 21 \).

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