General term of A.P. given `x+b,x+3b,x+5b,`Is…………..

Answer

To find the general term of the arithmetic progression (A.P.) given by the terms \(x + b\), \(x + 3b\), and \(x + 5b\), we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \(a\) of the A.P. is:
\[ a = x + b \]
To find the common difference \(d\), we subtract the first term from the second term:
\[ d = (x + 3b) - (x + b) = x + 3b - x - b = 2b \]

### Step 2: Write the formula for the nth term of an A.P.
The general formula for the nth term \(T_n\) of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]

### Step 3: Substitute the values of \(a\) and \(d\)
Now we substitute the values of \(a\) and \(d\) into the formula:
\[ T_n = (x + b) + (n - 1) \cdot (2b) \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ T_n = x + b + (n - 1) \cdot 2b \]
\[ T_n = x + b + 2bn - 2b \]
\[ T_n = x + 2bn - b \]
\[ T_n = x + 2bn - b \]

### Step 5: Final expression for the nth term
Thus, the general term of the A.P. is:
\[ T_n = x + 2bn - b \]