Answer:
a) u₂ = 7
b) k = 4 and r = 7
Step-by-step explanation:
A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Part (a)
Given recursive rule:
[tex]\begin{cases}u_{n+1}=\sqrt{2u_n+19}\\u_1=15\end{cases}[/tex]
To find u₂, substitute u₁ into the given equation:
[tex]\begin{aligned}u_2 & =\sqrt{2u_1+19}\\& = \sqrt{2(15)+19}\\& = \sqrt{30+19}\\& = \sqrt{49}\\& = 7\end{aligned}[/tex]
Part (b)
Given term-to-term rule:
[tex]u_{n+1}=ku_n+r[/tex]
(where k and r are constants)
Given terms of the arithmetic sequence:
Substitute the given terms into the equation to create two equations with r as the subject:
[tex]\begin{aligned}u_2 & = 179\\\implies ku_1+r & = 179\\k(43)+r & = 179\\r & = 179-43k\end{aligned}[/tex]
[tex]\begin{aligned}u_3 & = 723\\\implies ku_2+r & = 723\\k(179)+r & = 723\\r & = 723-179k\end{aligned}[/tex]
Substitute the second equation into the first and solve for k:
[tex]\begin{aligned}r & = r\\\implies723-179k & = 179-43k\\544-179k& = -43k\\544 & = 136k\\k & = \dfrac{544}{136}\\k & = 4\end{aligned}[/tex]
Substitute the found value of k into one of the previous equations and solve for r:
[tex]\begin{aligned}r & = 179-43k\\r & = 179-43(4)\\r & = 179-172\\r & = 7\end{aligned}[/tex]
Therefore, k = 4 and r = 7.
Substitute the found values of k and r to create the recursive rule for the given sequence:
[tex]\begin{cases}u_{n+1}=4u_n+7\\u_1=43\end{cases}[/tex]
Learn more about arithmetic sequences here:
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