Find the 10th term from 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 and common difference
The first term \( a \) of the A.P. is 4, and the second term is 9. The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 9 - 4 = 5 \]

### Step 2: Identify the last term
The last term of the A.P. is given as 254.

### Step 3: Reverse the A.P. for finding the 10th term from the end
To find the 10th term from the end, we can treat the sequence in reverse. In this case, the last term \( A \) is 254, and the common difference in the reverse A.P. will be:
\[ d' = a_1 - a_2 = 4 - 9 = -5 \]

### Step 4: Use the formula for the nth term
The formula for the nth term of an A.P. is given by:
\[ A_n = A + (n-1) \cdot d \]
For our case, we want to find the 10th term from the end, so \( n = 10 \):
\[ A_{10} = 254 + (10 - 1) \cdot (-5) \]

### Step 5: Calculate the 10th term
Now, substituting the values into the formula:
\[ A_{10} = 254 + 9 \cdot (-5) \]
\[ A_{10} = 254 - 45 \]
\[ A_{10} = 209 \]

### Conclusion
Thus, the 10th term from the end of the A.P. is **209**.

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