Each of the series `S_1 = 2+4+6+ ......... and S_2 = 3 +6+9+` .... is continued to 100 terms. Find how many terms are identical.

Options

A. 34
B. 33
C. 32
D. None of these

Answer

To solve the problem of finding how many terms are identical in the two series \( S_1 \) and \( S_2 \), we will follow these steps:

### Step 1: Identify the first series \( S_1 \)
The first series \( S_1 \) is given as:
\[ S_1 = 2 + 4 + 6 + \ldots \]
This is an arithmetic series where:
- First term \( a_1 = 2 \)
- Common difference \( d_1 = 2 \)

### Step 2: Find the last term of series \( S_1 \)
The formula for the last term of an arithmetic series is:
\[ a_n = a + (n-1) \cdot d \]
For \( S_1 \), with \( n = 100 \):
\[ a_{100} = 2 + (100 - 1) \cdot 2 = 2 + 198 = 200 \]
So, the last term of \( S_1 \) is 200.

### Step 3: Identify the second series \( S_2 \)
The second series \( S_2 \) is given as:
\[ S_2 = 3 + 6 + 9 + \ldots \]
This is also an arithmetic series where:
- First term \( a_2 = 3 \)
- Common difference \( d_2 = 3 \)

### Step 4: Find the last term of series \( S_2 \)
Using the same formula for the last term:
For \( S_2 \), with \( n = 100 \):
\[ a_{100} = 3 + (100 - 1) \cdot 3 = 3 + 297 = 300 \]
So, the last term of \( S_2 \) is 300.

### Step 5: Find the common terms in both series
To find the common terms, we need to find the least common multiple (LCM) of the common differences of the two series:
- \( d_1 = 2 \)
- \( d_2 = 3 \)

The LCM of 2 and 3 is 6.

### Step 6: Generate the common terms
The common terms will be multiples of 6:
\[ 6, 12, 18, \ldots \]

### Step 7: Find the last common term that is less than or equal to 200
The largest multiple of 6 that is less than or equal to 200 can be found by:
\[ 6n \leq 200 \]
\[ n \leq \frac{200}{6} \approx 33.33 \]
Thus, the largest integer \( n \) is 33.

### Step 8: Count the common terms
The common terms are:
\[ 6 \times 1, 6 \times 2, \ldots, 6 \times 33 \]
This gives us 33 common terms.

### Final Answer
The number of identical terms in the two series \( S_1 \) and \( S_2 \) is **33**.

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