Find the `11^(th)` term from the last term (towards the first term) of the A.P. : `10,7,4,"…",-62.`

Options

A. 32
B. -32
C. None of the above
D. 45

Answer

To find the 11th term from the last term towards the first term of the arithmetic progression (A.P.) given as \(10, 7, 4, \ldots, -62\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (A) and the common difference (D)**:
- The first term \(A\) is \(10\).
- The common difference \(D\) can be calculated as:
\[ D = 7 - 10 = -3 \]
- So, \(D = -3\).

2. **Identify the last term (L)**:
- The last term given in the sequence is \(-62\).

3. **Determine the number of terms (n) in the A.P.**:
- The formula for the \(n\)th term of an A.P. is given by:
\[ a_n = A + (n-1)D \]
- Setting \(a_n = -62\) (the last term), we can solve for \(n\):
\[ -62 = 10 + (n-1)(-3) \]
- Rearranging gives:
\[ -62 - 10 = (n-1)(-3) \implies -72 = (n-1)(-3) \]
- Dividing both sides by \(-3\):
\[ n - 1 = 24 \implies n = 25 \]

4. **Find the 11th term from the last term**:
- The 11th term from the last term is the \(n - 11 + 1\)th term from the first term, which is the \(15\)th term.
- Using the formula for the \(n\)th term again:
\[ a_{15} = A + (15-1)D \]
- Substituting the values:
\[ a_{15} = 10 + (14)(-3) \]
- Calculating:
\[ a_{15} = 10 - 42 = -32 \]

### Final Answer:
The 11th term from the last term towards the first term of the A.P. is \(-32\).

---