Find the `15^(th)` term of the AP, `x-7,x-2,x+3….`

Answer

To find the 15th term of the arithmetic progression (AP) given by the terms \(x - 7\), \(x - 2\), and \(x + 3\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(A\) of the AP is:
\[ A = x - 7 \]

To find the common difference \(D\), we can subtract the first term from the second term:
\[ D = (x - 2) - (x - 7) \]
Simplifying this:
\[ D = x - 2 - x + 7 = 5 \]

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \(A_n\) of an AP is given by:
\[ A_n = A + (n - 1) \cdot D \]
We need to find the 15th term, so we set \(n = 15\):
\[ A_{15} = A + (15 - 1) \cdot D \]

### Step 3: Substitute the values into the formula
Substituting the values we found:
\[ A_{15} = (x - 7) + (15 - 1) \cdot 5 \]
This simplifies to:
\[ A_{15} = (x - 7) + 14 \cdot 5 \]
Calculating \(14 \cdot 5\):
\[ 14 \cdot 5 = 70 \]
So we have:
\[ A_{15} = (x - 7) + 70 \]

### Step 4: Simplify the expression
Now, simplifying the expression:
\[ A_{15} = x - 7 + 70 = x + 63 \]

### Final Answer
Thus, the 15th term of the AP is:
\[ \boxed{x + 63} \]