Find the `8^(th)` term from the end of the A.P..`-12,-7,-2, …,68`

Answer

To find the 8th term from the end of the arithmetic progression (A.P.) given by the sequence: -12, -7, -2, ..., 68, we will follow these steps:

### Step 1: Identify the last term (L) and the common difference (D)
The last term of the A.P. is given as \( L = 68 \).

To find the common difference \( D \), we can subtract the first term from the second term:
\[ D = -7 - (-12) = -7 + 12 = 5 \]

### Step 2: Use the formula for the nth term from the end of an A.P.
The formula for the nth term from the end of an A.P. is:
\[ T_n = L - (n - 1) \cdot D \]
Where:
- \( T_n \) is the nth term from the end,
- \( L \) is the last term,
- \( n \) is the term number from the end,
- \( D \) is the common difference.

### Step 3: Substitute the values into the formula
We need to find the 8th term from the end, so we set \( n = 8 \):
\[ T_8 = 68 - (8 - 1) \cdot 5 \]
\[ T_8 = 68 - 7 \cdot 5 \]

### Step 4: Calculate the value
Now we calculate:
\[ T_8 = 68 - 35 \]
\[ T_8 = 33 \]

### Conclusion
The 8th term from the end of the A.P. is \( 33 \).

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