In the following two A.P. 's ow many terms are identical ?
`2,5,8,11…` to 60 terms and `3,5,7,…..50` terms

Answer

To find how many terms are identical in the two given arithmetic progressions (A.P.s), we will follow these steps:

### Step 1: Identify the first A.P. and its properties
The first A.P. is given as:
\[ 2, 5, 8, 11, \ldots \]
- First term \( a_1 = 2 \)
- Common difference \( d_1 = 5 - 2 = 3 \)
- Number of terms \( n_1 = 60 \)

The general term of the first A.P. can be expressed as:
\[ a_n = a_1 + (n-1)d_1 = 2 + (n-1) \cdot 3 = 3n - 1 \]

### Step 2: Identify the second A.P. and its properties
The second A.P. is given as:
\[ 3, 5, 7, \ldots \]
- First term \( a_2 = 3 \)
- Common difference \( d_2 = 5 - 3 = 2 \)
- Number of terms \( n_2 = 50 \)

The general term of the second A.P. can be expressed as:
\[ b_m = a_2 + (m-1)d_2 = 3 + (m-1) \cdot 2 = 2m + 1 \]

### Step 3: Set the general terms equal to find common terms
To find the common terms, we set the general terms equal to each other:
\[ 3n - 1 = 2m + 1 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 3n - 2m = 2 \]

### Step 5: Solve for \( n \) in terms of \( m \)
From the equation \( 3n - 2m = 2 \), we can express \( n \) in terms of \( m \):
\[ 3n = 2m + 2 \]
\[ n = \frac{2m + 2}{3} \]

### Step 6: Determine valid values of \( m \)
Since \( n \) must be a positive integer and \( n \) can take values from 1 to 60, we need \( 2m + 2 \) to be divisible by 3.

Let’s analyze the values of \( m \):
- \( m \) can take values from 1 to 50.
- For \( n \) to be an integer, \( 2m + 2 \) must be divisible by 3.

### Step 7: Check divisibility condition
We can check the values of \( m \):
- \( 2m + 2 \equiv 0 \mod 3 \)
- This simplifies to \( 2m \equiv 1 \mod 3 \)

The solutions for \( m \) modulo 3 are:
- \( m \equiv 2 \mod 3 \) (i.e., \( m = 2, 5, 8, \ldots, 50 \))

### Step 8: Count the valid \( m \) values
The sequence \( 2, 5, 8, \ldots, 50 \) is an arithmetic sequence where:
- First term \( = 2 \)
- Common difference \( = 3 \)

To find the number of terms:
Let \( k \) be the number of terms:
\[ 2 + (k-1) \cdot 3 = 50 \]
\[ (k-1) \cdot 3 = 48 \]
\[ k-1 = 16 \]
\[ k = 17 \]

### Conclusion
Thus, the number of identical terms in the two A.P.s is **17**.