Write the `8^(th)` term from the end of 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 can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is -12, and the last term \( l \) is 68.

To find the common difference \( d \):
- The second term is -7, so:
\[ d = -7 - (-12) = -7 + 12 = 5 \]

### Step 2: Determine the number of terms in the A.P.
To find the total number of terms \( n \) in the A.P., we can use the formula for the nth term of an A.P.:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 68 = -12 + (n-1) \cdot 5 \]
Now, solve for \( n \):
\[ 68 + 12 = (n-1) \cdot 5 \]
\[ 80 = (n-1) \cdot 5 \]
\[ n-1 = \frac{80}{5} = 16 \]
\[ n = 16 + 1 = 17 \]

### Step 3: Find the 8th term from the end
The 8th term from the end can be calculated using the formula for the kth term from the end:
\[ \text{Term from end} = a + (n-k)d \]
where \( k \) is the position from the end. Here, \( k = 8 \):
\[ \text{8th term from the end} = a + (n-8)d \]
Substituting the known values:
\[ = -12 + (17-8) \cdot 5 \]
\[ = -12 + 9 \cdot 5 \]
\[ = -12 + 45 \]
\[ = 33 \]

### Final Answer
The 8th term from the end of the A.P. is **33**.

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