1. **State the problem:** Given the arithmetic progression (AP) $x - 6, x - 1, x + 4, x + 9, \ldots$, find: (a) the first term $a$ and common difference $d$ (b) the general term $T(n)$ (c) the 8th term $T(8)$ 2. **Recall the formula for an arithmetic progression:** $$T(n) = a + (n - 1)d$$ where $a$ is the first term and $d$ is the common difference. 3. **Find the first term $a$ and common difference $d$: ** - The first term $a$ is the first element of the sequence: $a = x - 6$ - The common difference $d$ is the difference between consecutive terms: $$d = (x - 1) - (x - 6) = x - 1 - x + 6 = 5$$ 4. **Find the general term $T(n)$:** Using the formula: $$T(n) = a + (n - 1)d = (x - 6) + (n - 1) \times 5 = x - 6 + 5n - 5 = x + 5n - 11$$ 5. **Find the 8th term $T(8)$:** Substitute $n = 8$ into the general term: $$T(8) = x + 5 \times 8 - 11 = x + 40 - 11 = x + 29$$ **Final answers:** (a) $a = x - 6$, $d = 5$ (b) $T(n) = x + 5n - 11$ (c) $T(8) = x + 29$