Find the number of tterms in each of the following A.P.'s:
`18,15 (1)/(2), 13,…, - 47`

Answer

To find the number of terms in the given arithmetic progression (A.P.) `18, 15.5, 13, ..., -47`, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a_1 \) is 18.
- The second term \( a_2 \) is 15.5.
- The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 15.5 - 18 = -2.5 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a + (n-1) \cdot d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

### Step 3: Set up the equation for the last term
We need to find \( n \) such that the nth term \( a_n \) equals -47:
\[ -47 = 18 + (n-1)(-2.5) \]

### Step 4: Solve for n
1. Rearranging the equation:
\[ -47 - 18 = (n-1)(-2.5) \]
\[ -65 = (n-1)(-2.5) \]

2. Dividing both sides by -2.5:
\[ n - 1 = \frac{-65}{-2.5} \]
\[ n - 1 = 26 \]

3. Adding 1 to both sides:
\[ n = 26 + 1 = 27 \]

### Conclusion
The number of terms in the given A.P. is **27**.

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