Find the `10^(th)` term form the end of the A.P . 4 , 9, 14,`…..`,254 .

Answer

To find the 10th term from the end of the arithmetic progression (A.P.) 4, 9, 14, ..., 254, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term of the A.P. is:
\[ a = 4 \]
The common difference (d) can be calculated as:
\[ d = 9 - 4 = 5 \]

### Step 2: Identify the last term (l)
The last term of the A.P. is given as:
\[ l = 254 \]

### Step 3: Find the number of terms (n) in the A.P.
To find the number of terms in the A.P., we can use the formula for the nth term of an A.P.:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 254 = 4 + (n - 1) \cdot 5 \]
Now, simplify the equation:
\[ 254 - 4 = (n - 1) \cdot 5 \]
\[ 250 = (n - 1) \cdot 5 \]
Now, divide both sides by 5:
\[ n - 1 = 50 \]
So,
\[ n = 51 \]

### Step 4: Find the 10th term from the end
The 10th term from the end can be found using the formula for the nth term:
\[ A_k = a + (n - k) \cdot d \]
where \( k \) is the position from the end. Here, \( k = 10 \):
\[ A_{10} = a + (n - 10) \cdot d \]
Substituting the values we found:
\[ A_{10} = 4 + (51 - 10) \cdot 5 \]
\[ A_{10} = 4 + 41 \cdot 5 \]
\[ A_{10} = 4 + 205 \]
\[ A_{10} = 209 \]

### Final Answer
The 10th term from the end of the A.P. is:
\[ \boxed{209} \]