If \( m^{\text {th }} \) term of the sequence: \( 63,65,67,69, \ldots \ldots \ldots . \). and the sequence: \( 3,10,17,24, \ldots \ldots \ldots \ldots \) are equal then \( m=\ldots \ldots \ldots \ldots \) 12

The first sequence is an arithmetic sequence with the first term \( a_1 = 63 \) and a common difference \( d = 2 \). The \( m^{\text{th}} \) term can be calculated using the formula: \[ T_m = a_1 + (m-1)d = 63 + (m-1) \cdot 2 \] The second sequence also follows a pattern with the first term \( b_1 = 3 \) and a common difference of \( 7 \). The \( n^{\text{th}} \) term of this sequence is represented as: \[ U_n = b_1 + (n-1) \cdot 7 = 3 + (n-1) \cdot 7 \] To find when the two sequences are equal: \[ 63 + (m-1) \cdot 2 = 3 + (n-1) \cdot 7 \] If we were to assume \( n = m \) (i.e., we just want to find \( m \) such that both sequences yield the same term), we simplify to find \( m \): \[ 63 + (m-1) \cdot 2 = 3 + (m-1) \cdot 7 \] Expanding both sides results in: \[ 63 + 2m - 2 = 3 + 7m - 7 \] \[ 61 + 2m = 7m - 4 \] Rearranging gives: \[ 61 + 4 = 7m - 2m \] \[ 65 = 5m \] Thus, \[ m = \frac{65}{5} = 13 \] In conclusion, the value of \( m \) is \( 13 \).