To find the number of terms common to the two arithmetic progressions (APs) given, we can follow these steps:
### Step 1: Identify the first AP and its properties
The first arithmetic progression is:
\[ AP_1: 5, 9, 13, 17, \ldots, 216 \]
- First term (\(a_1\)) = 5
- Common difference (\(d_1\)) = 9 - 5 = 4
### Step 2: Identify the second AP and its properties
The second arithmetic progression is:
\[ AP_2: 3, 9, 15, 21, \ldots, 321 \]
- First term (\(a_2\)) = 3
- Common difference (\(d_2\)) = 9 - 3 = 6
### Step 3: Find the general term of both APs
The \(n\)-th term of an arithmetic progression can be expressed as:
\[ T_n = a + (n-1)d \]
For the first AP:
\[ T_{n1} = 5 + (n-1) \cdot 4 = 4n + 1 \]
For the second AP:
\[ T_{n2} = 3 + (m-1) \cdot 6 = 6m - 3 \]
### Step 4: Set the general terms equal to find common terms
To find common terms, we set the two general terms equal:
\[ 4n + 1 = 6m - 3 \]
### Step 5: Rearrange the equation
Rearranging gives us:
\[ 4n - 6m = -4 \]
Dividing the entire equation by 2 simplifies it:
\[ 2n - 3m = -2 \]
### Step 6: Express \(n\) in terms of \(m\)
From the equation \(2n - 3m = -2\):
\[ 2n = 3m - 2 \]
\[ n = \frac{3m - 2}{2} \]
### Step 7: Determine the values of \(m\) that yield integer \(n\)
For \(n\) to be an integer, \(3m - 2\) must be even. This means \(m\) must be even because \(3m\) is odd when \(m\) is odd, and subtracting 2 from an odd number gives an odd result.
Let \(m = 2k\) for integers \(k\):
\[ n = \frac{3(2k) - 2}{2} = \frac{6k - 2}{2} = 3k - 1 \]
### Step 8: Find the limits for \(m\) and \(n\)
Now we need to find the maximum values of \(n\) and \(m\) such that the terms are within the limits of the respective APs.
1. For \(AP_1\):
\[ 4n + 1 \leq 216 \]
\[ 4n \leq 215 \]
\[ n \leq 53.75 \]
Thus, \(n\) can take values up to 53.
2. For \(AP_2\):
\[ 6m - 3 \leq 321 \]
\[ 6m \leq 324 \]
\[ m \leq 54 \]
### Step 9: Determine the possible values of \(k\)
Since \(m = 2k\), we have:
\[ 2k \leq 54 \]
\[ k \leq 27 \]
### Step 10: Calculate the number of common terms
The values of \(k\) can range from 1 to 27 (since \(k\) must be a positive integer). Thus, the number of common terms is:
\[ k = 1, 2, 3, \ldots, 27 \]
This gives us a total of 27 common terms.
### Final Answer
The number of terms common to the two arithmetic progressions is **27**.
