1.8 Given the following sequience \( 12,15,18,21 \),.......... 2.8.1 Show that this is an arithmetic progression 1.8.2 Determine the \( 7^{\text {th }} \) term. 1.8.3 Determine the \( 24^{\text {th }} \) term.

To show that the sequence \( 12, 15, 18, 21, \ldots \) is an arithmetic progression, we need to check if the difference between consecutive terms is constant. The difference between the first term \( 15 - 12 = 3 \), the second term \( 18 - 15 = 3 \), and the third term \( 21 - 18 = 3 \) is the same, which confirms that this sequence is indeed an arithmetic progression with a common difference \( d = 3 \). To determine the \( 7^{\text{th}} \) term of the arithmetic progression, we can use the formula for the \( n^{\text{th}} \) term: \( a_n = a_1 + (n-1)d \). Here, \( a_1 = 12 \), \( d = 3 \), and \( n = 7 \). Plugging in the values: \[ a_7 = 12 + (7-1) \times 3 = 12 + 6 \times 3 = 12 + 18 = 30. \] To find the \( 24^{\text{th}} \) term, we use the same formula with \( n = 24 \): \[ a_{24} = 12 + (24-1) \times 3 = 12 + 23 \times 3 = 12 + 69 = 81. \] So, the \( 7^{\text{th}} \) term is \( 30 \), and the \( 24^{\text{th}} \) term is \( 81 \).