1 1~6 11 f) ) u_(1)+u_(3)+u_(5)=-12 u_(1)u_(2)u_(3)=8 1) u_(9)=5u_(2) và u_(13)=2u_(6)+5

The provided examples show how to solve for the first term (u1) and common difference (d) or common ratio (q) of arithmetic and geometric sequences, respectively, given various conditions. Let's tackle the problems you've presented:<br /><br /><br />**f) $\{ \begin{matrix} u_{1}+u_{3}+u_{5}=-12\\ u_{1}u_{2}u_{3}=8\end{matrix} $**<br /><br />This problem deals with an arithmetic sequence. We know that:<br /><br />* u_n = u₁ + (n-1)d (where u_n is the nth term, u₁ is the first term, and d is the common difference)<br /><br />Let's rewrite the equations using this formula:<br /><br />* u₁ + (u₁ + 2d) + (u₁ + 4d) = -12 => 3u₁ + 6d = -12 => u₁ + 2d = -4 (Equation 1)<br />* u₁(u₁ + d)(u₁ + 2d) = 8 (Equation 2)<br /><br />Substitute Equation 1 into Equation 2:<br /><br />* u₁(u₁ + d)(-4) = 8 => u₁(u₁ + d) = -2<br /><br />This gives us a system of two equations with two unknowns:<br /><br />* u₁ + 2d = -4<br />* u₁(u₁ + d) = -2<br /><br />Solving this system requires substitution or elimination. Let's solve for d in Equation 1: d = (-4 - u₁)/2<br /><br />Substitute this into the second equation:<br /><br />u₁(u₁ + (-4 - u₁)/2) = -2<br /><br />Solving this quadratic equation for u₁ will give you the first term. Once you have u₁, you can easily find d using Equation 1. This will likely involve some trial and error or using the quadratic formula.<br /><br /><br />**i) u₉ = 5u₂ and u₁₃ = 2u₆ + 5**<br /><br />This problem deals with an arithmetic sequence. Using the formula u_n = u₁ + (n-1)d:<br /><br />* u₉ = u₁ + 8d = 5(u₁ + d) => u₁ + 8d = 5u₁ + 5d => 4u₁ - 3d = 0 (Equation 1)<br />* u₁₃ = u₁ + 12d = 2(u₁ + 5d) + 5 => u₁ + 12d = 2u₁ + 10d + 5 => u₁ - 2d = -5 (Equation 2)<br /><br />Now we have a system of two linear equations:<br /><br />* 4u₁ - 3d = 0<br />* u₁ - 2d = -5<br /><br />Solve this system (e.g., using substitution or elimination) to find u₁ and d. The easiest approach here is to solve Equation 2 for u₁ (u₁ = 2d - 5) and substitute into Equation 1.<br /><br /><br />Remember to check your solutions by plugging them back into the original equations to verify they are correct. The solutions will be integers based on the examples provided.<br />