1. Problem 1: Given the arithmetic sequence $(2m-5, 4, 9)$, find $m$ and determine if the sequence is increasing or decreasing. 2. Recall that in an arithmetic sequence, the difference between consecutive terms is constant: $$a_2 - a_1 = a_3 - a_2$$ 3. Substitute the terms: $$4 - (2m - 5) = 9 - 4$$ 4. Simplify both sides: $$4 - 2m + 5 = 5$$ $$9 - 4 = 5$$ 5. So: $$9 - 2m = 5$$ 6. Solve for $m$: $$9 - 2m = 5$$ $$\Rightarrow -2m = 5 - 9$$ $$\Rightarrow -2m = -4$$ $$\Rightarrow m = \frac{\cancel{-4}}{\cancel{-2}} = 2$$ 7. To check if the sequence is increasing or decreasing, find the common difference: $$d = a_2 - a_1 = 4 - (2\times 2 - 5) = 4 - (4 - 5) = 4 - (-1) = 5 > 0$$ 8. Since $d > 0$, the sequence is increasing. --- 9. Problem 2: Given arithmetic sequence $(a_n)$ with $a_1=7$ and $a_2=13$, find $a_{10}$. 10. The common difference is: $$d = a_2 - a_1 = 13 - 7 = 6$$ 11. The formula for the $n$-th term of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ 12. Substitute $n=10$: $$a_{10} = 7 + (10-1) \times 6 = 7 + 9 \times 6 = 7 + 54 = 61$$ --- 13. Problem 3: Given sequences $( -1, 2, x )$ arithmetic and $( -1, 2, y )$ geometric, find conditions on $x$ and $y$. 14. For arithmetic sequence: $$2 - (-1) = x - 2$$ $$3 = x - 2$$ $$x = 5$$ 15. For geometric sequence: $$\frac{2}{-1} = \frac{y}{2}$$ $$-2 = \frac{y}{2}$$ $$y = -4$$ 16. So $x=5 > 0$ and $y = -4 < 0$. 17. The correct condition is: $$x > 0 \text{ and } y < 0$$ Final answers: 1. $m=2$, sequence is increasing. 2. $a_{10} = 61$. 3. $x > 0$ and $y < 0$.