What is the 10th common term between the series `2+6+10+....."and "1+6+11+.....?`

Options

A. 180
B. 186
C. 196
D. 206

Explanation

To find the 10th common term between the two series \(2, 6, 10, \ldots\) and \(1, 6, 11, \ldots\), we will follow these steps:

### Step 1: Identify the general term of the first series
The first series is \(2, 6, 10, \ldots\). This is an arithmetic series where:
- The first term \(a_1 = 2\)
- The common difference \(d_1 = 4\)

The \(n\)-th term of this series can be expressed as:
\[ T_n = a_1 + (n-1)d_1 = 2 + (n-1) \cdot 4 = 4n - 2 \]

### Step 2: Identify the general term of the second series
The second series is \(1, 6, 11, \ldots\). This is also an arithmetic series where:
- The first term \(b_1 = 1\)
- The common difference \(d_2 = 5\)

The \(m\)-th term of this series can be expressed as:
\[ S_m = b_1 + (m-1)d_2 = 1 + (m-1) \cdot 5 = 5m - 4 \]

### Step 3: Find common terms
To find the common terms, we need to set the \(n\)-th term of the first series equal to the \(m\)-th term of the second series:
\[ 4n - 2 = 5m - 4 \]
Rearranging gives:
\[ 4n - 5m = -2 \]

### Step 4: Solve for integer solutions
We can rewrite the equation as:
\[ 4n = 5m - 2 \]
This means \(5m - 2\) must be divisible by 4. We can express \(m\) in terms of \(n\):
\[ 5m = 4n + 2 \implies m = \frac{4n + 2}{5} \]
For \(m\) to be an integer, \(4n + 2\) must be divisible by 5.

### Step 5: Find values of \(n\) that satisfy the divisibility condition
We can check values of \(n\):
- For \(n = 1\): \(4(1) + 2 = 6\) (not divisible by 5)
- For \(n = 2\): \(4(2) + 2 = 10\) (divisible by 5) → \(m = 2\)
- For \(n = 3\): \(4(3) + 2 = 14\) (not divisible by 5)
- For \(n = 4\): \(4(4) + 2 = 18\) (not divisible by 5)
- For \(n = 5\): \(4(5) + 2 = 22\) (not divisible by 5)
- For \(n = 6\): \(4(6) + 2 = 26\) (divisible by 5) → \(m = 6\)
- For \(n = 7\): \(4(7) + 2 = 30\) (not divisible by 5)
- For \(n = 8\): \(4(8) + 2 = 34\) (not divisible by 5)
- For \(n = 9\): \(4(9) + 2 = 38\) (not divisible by 5)
- For \(n = 10\): \(4(10) + 2 = 42\) (divisible by 5) → \(m = 8\)

Continuing this process, we can find the common terms:
1. \(n = 2\) → Common term = 6
2. \(n = 6\) → Common term = 26
3. \(n = 10\) → Common term = 46
4. \(n = 14\) → Common term = 66
5. \(n = 18\) → Common term = 86
6. \(n = 22\) → Common term = 106
7. \(n = 26\) → Common term = 126
8. \(n = 30\) → Common term = 146
9. \(n = 34\) → Common term = 166
10. \(n = 38\) → Common term = 186

### Step 6: Conclusion
The 10th common term between the two series is:
\[ \boxed{186} \]