Find how many terms are there in the A.P . 16 , 24, 32, `….` 96

Answer

To find how many terms are there in the arithmetic progression (A.P.) given by 16, 24, 32, ..., 96, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) is given as:
\[ a = 16 \]
To find the common difference \( d \), we subtract the first term from the second term:
\[ d = a_2 - a_1 = 24 - 16 = 8 \]

### Step 2: Identify the last term (l)
The last term \( l \) is given as:
\[ l = 96 \]

### Step 3: Use the formula for the nth term of an A.P.
The formula for the nth term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
We know \( a_n = 96 \), \( a = 16 \), and \( d = 8 \). Plugging these values into the formula gives:
\[ 96 = 16 + (n - 1) \cdot 8 \]

### Step 4: Solve for n
First, subtract 16 from both sides:
\[ 96 - 16 = (n - 1) \cdot 8 \]
\[ 80 = (n - 1) \cdot 8 \]
Next, divide both sides by 8:
\[ 10 = n - 1 \]
Now, add 1 to both sides to find \( n \):
\[ n = 10 + 1 = 11 \]

### Conclusion
The total number of terms in the A.P. is:
\[ \boxed{11} \]

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