1. **State the problem:** Write recursive and explicit equations for the given sequences. 2. **Part (a): Sequence 108, 120, 132, ...** - This is an arithmetic sequence where each term increases by 12. - Recursive formula: $a_1 = 108$, $a_n = a_{n-1} + 12$ for $n > 1$. - Explicit formula: $t(n) = 108 + (n-1) \times 12 = 12n + 96$. 3. **Part (b): Sequence 2/5, 4/5, 8/5, ...** - This is a geometric sequence where each term is multiplied by 2. - Recursive formula: $a_1 = \frac{2}{5}$, $a_n = 2 \times a_{n-1}$ for $n > 1$. - Explicit formula: $t(n) = \frac{2}{5} \times 2^{n-1} = \frac{2^n}{5}$. 4. **Part (c): Sequence 3741, 3702, 3663, ...** - This is an arithmetic sequence decreasing by 39 each time. - Recursive formula: $a_1 = 3741$, $a_n = a_{n-1} - 39$ for $n > 1$. - Explicit formula: $t(n) = 3741 - 39(n-1) = 3780 - 39n$. 5. **Part (d): Sequence 117, 23.4, 4.68, ...** - This is a geometric sequence where each term is multiplied by 0.2. - Recursive formula: $a_1 = 117$, $a_n = 0.2 \times a_{n-1}$ for $n > 1$. - Explicit formula: $t(n) = 117 \times 0.2^{n-1}$. **Final answers:** - (a) Recursive: $a_n = a_{n-1} + 12$, Explicit: $t(n) = 12n + 96$ - (b) Recursive: $a_n = 2a_{n-1}$, Explicit: $t(n) = \frac{2^n}{5}$ - (c) Recursive: $a_n = a_{n-1} - 39$, Explicit: $t(n) = 3780 - 39n$ - (d) Recursive: $a_n = 0.2a_{n-1}$, Explicit: $t(n) = 117 \times 0.2^{n-1}$