4 For each of the following arithmetic sequences, calculate: i the value of \( p \) ii the sequence iii the value of the 25 th term. a \( p+4 ; 3 p+1 ; 4 p \)

To find the value of \( p \), we need to ensure that the difference between consecutive terms is constant. 1. The difference between the second and first term: \[ (3p + 1) - (p + 4) = 2p - 3 \] 2. The difference between the third and second term: \[ (4p) - (3p + 1) = p - 1 \] Setting these differences equal gives us: \[ 2p - 3 = p - 1 \] Solving for \( p \): \[ 2p - p = -1 + 3 \implies p = 2 \] Now substituting \( p \) back into the sequence: - First term: \( 2 + 4 = 6 \) - Second term: \( 3(2) + 1 = 7 \) - Third term: \( 4(2) = 8 \) Thus, the sequence is \( 6, 7, 8, \ldots \) which is an arithmetic sequence with a common difference of \( 1 \). To find the 25th term, we use the general formula for the nth term in an arithmetic sequence: \[ a_n = a_1 + (n - 1)d \] Here, \( a_1 = 6 \), \( d = 1 \), and \( n = 25 \): \[ a_{25} = 6 + (25 - 1) \cdot 1 = 6 + 24 = 30 \] So, the value of \( p \) is \( 2 \), the sequence is \( 6, 7, 8, \ldots \), and the value of the 25th term is \( 30 \).