A sequence is generated by the rule that the xth is `x^2+1` for each positive integer x. In this sequence for any value `x gt 1` the value of `(x+1)th` term less the value of xth term is

Options

A. A)`2x^2+1`
B. B)`x^2+1`
C. C)`2x+1`
D. D)`x+2`

Answer

To solve the problem, we need to find the difference between the \((x+1)\)th term and the \(x\)th term of the sequence defined by the rule \(a_x = x^2 + 1\).

### Step-by-Step Solution:

1. **Identify the \(x\)th term**:
The \(x\)th term of the sequence is given by:
\[ a_x = x^2 + 1 \]

2. **Identify the \((x+1)\)th term**:
The \((x+1)\)th term of the sequence is given by substituting \(x\) with \(x+1\):
\[ a_{x+1} = (x+1)^2 + 1 \]

3. **Expand the \((x+1)\)th term**:
Now, we need to expand \((x+1)^2 + 1\):
\[ a_{x+1} = (x^2 + 2x + 1) + 1 = x^2 + 2x + 2 \]

4. **Calculate the difference**:
We need to find the difference between the \((x+1)\)th term and the \(x\)th term:
\[ a_{x+1} - a_x = (x^2 + 2x + 2) - (x^2 + 1) \]

5. **Simplify the expression**:
Now, simplify the expression:
\[ a_{x+1} - a_x = x^2 + 2x + 2 - x^2 - 1 \]
The \(x^2\) terms cancel out:
\[ = 2x + 2 - 1 = 2x + 1 \]

6. **Final Result**:
Thus, the value of \((x+1)\)th term minus \(x\)th term is:
\[ a_{x+1} - a_x = 2x + 1 \]