The largest term common to the sequences `1, 11 , 21 , 31 , to100` terms and `31 , 36 , 41 , 46 , to100` terms is `381` b. `471` c. `281` d. none of these

Options

A. 381
B. 471
C. 281
D. 521

Explanation

To find the largest term common to the sequences \(1, 11, 21, 31, \ldots\) and \(31, 36, 41, 46, \ldots\), we will follow these steps:

### Step 1: Identify the first arithmetic progression (AP)
The first sequence is:
- First term \(a_1 = 1\)
- Common difference \(d_1 = 10\)

The \(n\)-th term of an AP can be calculated using the formula:
\[ T_n = a + (n-1)d \]
For the first AP, the \(n\)-th term is:
\[ T_n = 1 + (n-1) \cdot 10 = 10n - 9 \]

### Step 2: Identify the second arithmetic progression (AP)
The second sequence is:
- First term \(a_2 = 31\)
- Common difference \(d_2 = 5\)

The \(m\)-th term of this AP is:
\[ T_m = 31 + (m-1) \cdot 5 = 5m + 26 \]

### Step 3: Set the two sequences equal to find common terms
To find the common terms, we set the two expressions equal:
\[ 10n - 9 = 5m + 26 \]
Rearranging gives:
\[ 10n - 5m = 35 \quad \text{(1)} \]

### Step 4: Solve for integer values of \(n\) and \(m\)
From equation (1), we can express \(m\) in terms of \(n\):
\[ 5m = 10n - 35 \implies m = 2n - 7 \]
Since \(m\) must be a positive integer, we require:
\[ 2n - 7 \geq 1 \implies 2n \geq 8 \implies n \geq 4 \]

### Step 5: Find the maximum \(n\) such that \(T_n\) is within the limits of the first AP
The first AP has 100 terms, so \(n\) can take values from 1 to 100. The maximum value of \(n\) satisfying \(2n - 7 \geq 1\) is:
\[ n \leq 100 \]

### Step 6: Calculate the corresponding \(m\) and find the common term
We will find the common term for the maximum value of \(n\):
- For \(n = 100\):
\[ m = 2(100) - 7 = 193 \quad \text{(not valid since \(m\) must be ≤ 100)} \]
- For \(n = 99\):
\[ m = 2(99) - 7 = 191 \quad \text{(not valid)} \]
- Continue this until \(n = 8\):
\[ m = 2(8) - 7 = 9 \quad \text{(valid)} \]

### Step 7: Find the common term for \(n = 8\)
Now, calculate the common term:
\[ T_8 = 10(8) - 9 = 71 \]
\[ T_9 = 5(9) + 26 = 71 \quad \text{(both give the same term)} \]

### Step 8: Check higher values of \(n\) until reaching the limit
Continue checking until \(n = 53\):
\[ m = 2(53) - 7 = 99 \quad \text{(valid)} \]
Calculate:
\[ T_{53} = 10(53) - 9 = 521 \]

### Conclusion
The largest term common to both sequences is \(521\). Since this value is not in the options provided, the answer is:
**d. none of these**