1. **State the problem:** Find the first two terms of an arithmetic sequence given that the sixth term is 21 and the sum of the first seventeen terms is 0. 2. **Recall formulas for arithmetic sequences:** - The $n$th term is given by $$u_n = a + (n-1)d$$ where $a$ is the first term and $d$ is the common difference. - The sum of the first $n$ terms is $$S_n = \frac{n}{2} (2a + (n-1)d)$$. 3. **Use the given information:** - Sixth term: $$u_6 = a + 5d = 21$$ - Sum of first 17 terms: $$S_{17} = \frac{17}{2} (2a + 16d) = 0$$ 4. **Write the equations:** $$a + 5d = 21 \quad (1)$$ $$\frac{17}{2} (2a + 16d) = 0 \implies 17(a + 8d) = 0 \implies a + 8d = 0 \quad (2)$$ 5. **Solve the system:** From (2): $$a = -8d$$ Substitute into (1): $$-8d + 5d = 21$$ $$\cancel{-8d} + 5d = 21$$ $$-3d = 21$$ $$d = -7$$ 6. **Find $a$:** $$a = -8(-7) = 56$$ 7. **Answer:** The first term is $a = 56$ and the common difference is $d = -7$. 8. **Check:** Sixth term: $$u_6 = 56 + 5(-7) = 56 - 35 = 21$$ correct. Sum of first 17 terms: $$S_{17} = \frac{17}{2} (2 \times 56 + 16 \times (-7)) = \frac{17}{2} (112 - 112) = 0$$ correct.