To find the number of identical terms in the two given arithmetic progressions (A.P.s), we will follow these steps:
### Step 1: Identify the first A.P.
The first A.P. is given as:
- First term (a₁) = 2
- Common difference (d₁) = 3
- Number of terms (n₁) = 60
The nth term of an A.P. can be calculated using the formula:
\[ T_n = a + (n-1) \cdot d \]
For the first A.P.:
\[ T_{60} = 2 + (60-1) \cdot 3 \]
\[ T_{60} = 2 + 59 \cdot 3 \]
\[ T_{60} = 2 + 177 \]
\[ T_{60} = 179 \]
So, the first A.P. is:
\[ 2, 5, 8, 11, \ldots, 179 \]
### Step 2: Identify the second A.P.
The second A.P. is given as:
- First term (a₂) = 3
- Common difference (d₂) = 2
- Number of terms (n₂) = 50
Using the same formula for the second A.P.:
\[ T_{50} = 3 + (50-1) \cdot 2 \]
\[ T_{50} = 3 + 49 \cdot 2 \]
\[ T_{50} = 3 + 98 \]
\[ T_{50} = 101 \]
So, the second A.P. is:
\[ 3, 5, 7, 9, \ldots, 101 \]
### Step 3: Find the common terms in both A.P.s
Now, we need to find the common terms between the two A.P.s.
The first A.P. is:
\[ 2, 5, 8, 11, \ldots, 179 \]
The second A.P. is:
\[ 3, 5, 7, 9, \ldots, 101 \]
### Step 4: Identify the range of common terms
The common terms will be those that appear in both sequences. The first common term is 5.
To find the last common term, we need to find the largest term in the first A.P. that is less than or equal to 101 (the last term of the second A.P.).
The general term of the first A.P. can be expressed as:
\[ T_n = 2 + (n-1) \cdot 3 \]
Setting this equal to the last term of the second A.P. (101):
\[ 2 + (n-1) \cdot 3 = 101 \]
\[ (n-1) \cdot 3 = 99 \]
\[ n-1 = 33 \]
\[ n = 34 \]
So, the common terms in the first A.P. are:
\[ 5, 11, 17, \ldots, 101 \]
### Step 5: Find the number of common terms
The common terms form another A.P. with:
- First term = 5
- Common difference = 6 (since the next common term after 5 is 11, which is 5 + 6)
- Last term = 101
Using the formula for the nth term:
\[ T_n = 5 + (n-1) \cdot 6 \]
Setting this equal to 101:
\[ 5 + (n-1) \cdot 6 = 101 \]
\[ (n-1) \cdot 6 = 96 \]
\[ n-1 = 16 \]
\[ n = 17 \]
Thus, the number of identical terms in both A.P.s is **17**.
