210 is \( 51^{\text {st }} \) term of the sequence \( 10,15,20,25, \ldots \)

To determine if 210 is the \( 51^{st} \) term of the sequence \( 10, 15, 20, 25, \ldots \), we first need to identify the general formula for the \( n^{th} \) term of this arithmetic sequence. 1. **Identify the first term and common difference:** - The first term \( a = 10 \) - The common difference \( d = 15 - 10 = 5 \) 2. **Write the formula for the \( n^{th} \) term of an arithmetic sequence:** The formula for the \( n^{th} \) term \( T_n \) is given by: \[ T_n = a + (n - 1) \cdot d \] 3. **Substitute the known values into the formula:** For \( n = 51 \): \[ T_{51} = 10 + (51 - 1) \cdot 5 \] 4. **Calculate \( T_{51} \):** \[ T_{51} = 10 + 50 \cdot 5 \] \[ T_{51} = 10 + 250 \] \[ T_{51} = 260 \] Now we can conclude that the \( 51^{st} \) term of the sequence is 260, not 210. Therefore, 210 is not the \( 51^{st} \) term of the sequence.