How many terms are there in the AP 18, `15(1)/(2), 13,…, -47?`

Answer

To find how many terms are in the arithmetic progression (AP) 18, \(15\frac{1}{2}\), 13, ..., -47, we will follow these steps:

### Step 1: Identify the first term (a) and the last term (l)
The first term \(a\) is 18, and the last term \(l\) is -47.

### Step 2: Find the common difference (d)
To find the common difference \(d\), we can subtract the first term from the second term:
\[ d = 15\frac{1}{2} - 18 \]
Convert \(15\frac{1}{2}\) to an improper fraction:
\[ 15\frac{1}{2} = \frac{31}{2} \]
Now calculate \(d\):
\[ d = \frac{31}{2} - 18 = \frac{31}{2} - \frac{36}{2} = \frac{31 - 36}{2} = \frac{-5}{2} \]

### Step 3: Use the formula for the nth term of an AP
The nth term of an AP can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
We know \(a_n = -47\), \(a = 18\), and \(d = -\frac{5}{2}\). Plugging in these values:
\[ -47 = 18 + (n - 1) \cdot \left(-\frac{5}{2}\right) \]

### Step 4: Solve for n
Rearranging the equation:
\[ -47 - 18 = (n - 1) \cdot \left(-\frac{5}{2}\right) \]
\[ -65 = (n - 1) \cdot \left(-\frac{5}{2}\right) \]
Multiply both sides by \(-\frac{2}{5}\):
\[ n - 1 = \frac{-65 \cdot -2}{5} = \frac{130}{5} = 26 \]
Now, add 1 to both sides to find \(n\):
\[ n = 26 + 1 = 27 \]

### Conclusion
There are **27 terms** in the arithmetic progression.

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