The 10th common term between the series 3+7+11+ . . . And 1+6+11+ . . . ., is

Options

A. 191
B. 193
C. 211
D. none of these

Explanation

To find the 10th common term between the two arithmetic series \(3, 7, 11, \ldots\) and \(1, 6, 11, \ldots\), we will follow these steps:

### Step 1: Identify the first terms and common differences of both series.

- The first series is \(3, 7, 11, \ldots\)
- First term \(a_1 = 3\)
- Common difference \(d_1 = 7 - 3 = 4\)

- The second series is \(1, 6, 11, \ldots\)
- First term \(a_2 = 1\)
- Common difference \(d_2 = 6 - 1 = 5\)

### Step 2: Write the general term for both series.

- The \(n\)th term of the first series can be expressed as:
\[ T_n^{(1)} = a_1 + (n-1)d_1 = 3 + (n-1) \cdot 4 = 4n - 1 \]

- The \(n\)th term of the second series can be expressed as:
\[ T_n^{(2)} = a_2 + (n-1)d_2 = 1 + (n-1) \cdot 5 = 5n - 4 \]

### Step 3: Set the two general terms equal to find common terms.

To find the common terms, we set the two expressions equal:
\[ 4n - 1 = 5m - 4 \]
Rearranging gives:
\[ 4n - 5m = -3 \quad \text{(1)} \]

### Step 4: Solve for integer solutions.

We need to find integer solutions for the equation \(4n - 5m = -3\). We can express \(n\) in terms of \(m\):
\[ 4n = 5m - 3 \implies n = \frac{5m - 3}{4} \]
For \(n\) to be an integer, \(5m - 3\) must be divisible by \(4\).

### Step 5: Find values of \(m\) that satisfy the divisibility condition.

We can check values of \(m\):
- For \(m = 1\): \(5(1) - 3 = 2\) (not divisible by 4)
- For \(m = 2\): \(5(2) - 3 = 7\) (not divisible by 4)
- For \(m = 3\): \(5(3) - 3 = 12\) (divisible by 4, \(n = 3\))
- For \(m = 4\): \(5(4) - 3 = 17\) (not divisible by 4)
- For \(m = 5\): \(5(5) - 3 = 22\) (divisible by 4, \(n = 5\))

Continuing this process, we find:
- \(m = 8\) gives \(n = 10\)
- \(m = 13\) gives \(n = 15\)
- ...

### Step 6: Find the 10th common term.

The common terms occur at \(m = 3, 5, 8, 10, 13, \ldots\). The pattern shows that \(m\) can be expressed as:
\[ m = 3 + 5k \quad \text{for } k = 0, 1, 2, \ldots \]
Thus, the \(k\)th common term corresponds to \(m = 3 + 5(k-1)\).

For \(k = 10\):
\[ m = 3 + 5(10 - 1) = 3 + 45 = 48 \]

### Step 7: Calculate the 10th common term.

Substituting \(m = 48\) into the second series:
\[ T_{48}^{(2)} = 5(48) - 4 = 240 - 4 = 236 \]

Thus, the 10th common term between the two series is **236**.