1. **Stating the problem:** We have a series with terms $y$ corresponding to $x$ values. The 5th term is given as 12.5. We need to find the 1st and 10th terms of the series. 2. **Given data:** $x$: 3, 4, 5, 6, 7, 8, 9 $y$: 2.7, 6.4, 12.5, 21.6, 34.3, 51.2, 72.9 3. **Observing the series:** The terms do not form an arithmetic or geometric progression. We will try to find a polynomial relation between $x$ and $y$. 4. **Assuming a quadratic polynomial:** $$y = ax^2 + bx + c$$ 5. **Using three points to find $a$, $b$, and $c$:** Using points $(3, 2.7)$, $(5, 12.5)$, and $(7, 34.3)$: From $(3, 2.7)$: $$9a + 3b + c = 2.7$$ From $(5, 12.5)$: $$25a + 5b + c = 12.5$$ From $(7, 34.3)$: $$49a + 7b + c = 34.3$$ 6. **Solving the system:** Subtract first from second: $$(25a - 9a) + (5b - 3b) + (c - c) = 12.5 - 2.7$$ $$16a + 2b = 9.8$$ Subtract second from third: $$(49a - 25a) + (7b - 5b) + (c - c) = 34.3 - 12.5$$ $$24a + 2b = 21.8$$ 7. **Subtract the two equations:** $$(24a + 2b) - (16a + 2b) = 21.8 - 9.8$$ $$8a = 12$$ $$a = 1.5$$ 8. **Find $b$:** From $16a + 2b = 9.8$: $$16(1.5) + 2b = 9.8$$ $$24 + 2b = 9.8$$ $$2b = 9.8 - 24 = -14.2$$ $$b = -7.1$$ 9. **Find $c$:** From $9a + 3b + c = 2.7$: $$9(1.5) + 3(-7.1) + c = 2.7$$ $$13.5 - 21.3 + c = 2.7$$ $$c = 2.7 + 7.8 = 10.5$$ 10. **Polynomial found:** $$y = 1.5x^2 - 7.1x + 10.5$$ 11. **Find the 1st term ($x=1$):** $$y_1 = 1.5(1)^2 - 7.1(1) + 10.5 = 1.5 - 7.1 + 10.5 = 4.9$$ 12. **Find the 10th term ($x=10$):** $$y_{10} = 1.5(10)^2 - 7.1(10) + 10.5 = 1.5(100) - 71 + 10.5 = 150 - 71 + 10.5 = 89.5$$ **Final answer:** - First term: $4.9$ - Tenth term: $89.5$