Find the 11th term from the last term (towards the first term) of the AP 12, 8, 4,... - 84.

Answer

To find the 11th term from the last term towards the first term of the arithmetic progression (AP) given as 12, 8, 4, ..., -84, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \( A \) of the AP is 12. The common difference \( D \) can be calculated as:
\[ D = 8 - 12 = -4 \]

### Step 2: Identify the last term (L)
The last term given in the AP is -84.

### Step 3: Determine the number of terms (n) in the AP
To find the number of terms in the AP, we can use the formula for the nth term of an AP:
\[ L = A + (n - 1)D \]
Substituting the known values:
\[ -84 = 12 + (n - 1)(-4) \]
Rearranging the equation:
\[ -84 - 12 = (n - 1)(-4) \]
\[ -96 = (n - 1)(-4) \]
Dividing both sides by -4:
\[ n - 1 = 24 \]
Adding 1 to both sides gives:
\[ n = 25 \]

### Step 4: Find the 11th term from the last term
To find the 11th term from the last term, we can use the formula for the nth term:
\[ A_n = A + (n - 1)D \]
In this case, we want the 11th term from the last, which means we need to find the 15th term of the original sequence (since \( 25 - 11 + 1 = 15 \)):
\[ A_{15} = A + (15 - 1)D \]
Substituting the values:
\[ A_{15} = 12 + (14)(-4) \]
Calculating:
\[ A_{15} = 12 - 56 = -44 \]

### Final Answer
The 11th term from the last term towards the first term of the AP is \(-44\).

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