The 10th common terms between the series `3+7+11+`….. And `1+6+11+`….. is
(i) 191
(ii) 193
(iii) 211
(iv) None of these

Options

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

Answer

To find the 10th common term between the two series, we first need to identify the general terms of each series.

### Step 1: Identify the first series
The first series is:
3, 7, 11, 15, 19, ...
This is an arithmetic progression (AP) where:
- First term (a₁) = 3
- Common difference (d₁) = 7 - 3 = 4

The nth term of an arithmetic series can be given by the formula:
\[ a_n = a_1 + (n - 1) \cdot d \]
For the first series:
\[ a_n = 3 + (n - 1) \cdot 4 \]
So, the nth term is:
\[ a_n = 4n - 1 \]

### Step 2: Identify the second series
The second series is:
1, 6, 11, 16, 21, ...
This is also an arithmetic progression where:
- First term (b₁) = 1
- Common difference (d₂) = 6 - 1 = 5

Using the same formula for the nth term:
\[ b_n = 1 + (n - 1) \cdot 5 \]
So, the nth term is:
\[ b_n = 5n - 4 \]

### Step 3: Find common terms
To find the common terms, we need to set the two nth terms equal to each other:
\[ 4m - 1 = 5n - 4 \]
Rearranging gives:
\[ 4m - 5n = -3 \]

### Step 4: Solve for integer solutions
We can rearrange this equation:
\[ 4m = 5n - 3 \]
\[ m = \frac{5n - 3}{4} \]

For m to be an integer, \( 5n - 3 \) must be divisible by 4. We can check values of n to find integer values of m.

### Step 5: Find values of n
Let’s check for n = 1, 2, 3, ... until we find the 10th common term:

1. n = 1: \( 5(1) - 3 = 2 \) (not divisible by 4)
2. n = 2: \( 5(2) - 3 = 7 \) (not divisible by 4)
3. n = 3: \( 5(3) - 3 = 12 \) (divisible by 4, m = 3)
4. n = 4: \( 5(4) - 3 = 17 \) (not divisible by 4)
5. n = 5: \( 5(5) - 3 = 22 \) (not divisible by 4)
6. n = 6: \( 5(6) - 3 = 27 \) (not divisible by 4)
7. n = 7: \( 5(7) - 3 = 32 \) (divisible by 4, m = 8)
8. n = 8: \( 5(8) - 3 = 37 \) (not divisible by 4)
9. n = 9: \( 5(9) - 3 = 42 \) (not divisible by 4)
10. n = 10: \( 5(10) - 3 = 47 \) (not divisible by 4)
11. n = 11: \( 5(11) - 3 = 52 \) (divisible by 4, m = 13)
12. n = 12: \( 5(12) - 3 = 57 \) (not divisible by 4)
13. n = 13: \( 5(13) - 3 = 62 \) (not divisible by 4)
14. n = 14: \( 5(14) - 3 = 67 \) (not divisible by 4)
15. n = 15: \( 5(15) - 3 = 72 \) (divisible by 4, m = 18)

Continuing this process, we find the common terms:
- 1st common term: 11
- 2nd common term: 31
- 3rd common term: 51
- 4th common term: 71
- 5th common term: 91
- 6th common term: 111
- 7th common term: 131
- 8th common term: 151
- 9th common term: 171
- 10th common term: 191

### Step 6: Conclusion
The 10th common term is:
\[ \text{10th common term} = 191 \]

### Final Answer
Thus, the answer is option (i) 191.

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