Question

1. given: l = √2 m = 3√2 n = √16 p = √9

circle the expressions that will result in a rational and cross out the expressions that will result in an irrational.
l + m n + p p + l m + p
l · m n · p p · l m · p

1. given the function aₙ = 3 - 4(n - 1), determine the following.

recursive form →

1. determine the recursive form and explicit form for the following sequence.

112, 107, 102, 97, ...
recursive form → explicit form →

1. the linear function f(x) = -3x + 2, where x is a natural number, defines a sequence. what is the first term of the sequence? write the recursive formula that defines the same sequence.

a₁ → recursive form →

1. which point(s) are solutions to the system graphed below?

a. (-2, -3)
b. (-1, 2)
c. (2, 1)
d. (-4,0)

Explanation:

Step1: Simplify given values

$L = \sqrt{2}$, $M = 3\sqrt{2}$, $N=\sqrt{16}=4$, $P=\sqrt{9}=3$

Step2: Analyze sums

• $L + M=\sqrt{2}+3\sqrt{2}=4\sqrt{2}$ (irrational).
• $N + P=4 + 3=7$ (rational).
• $P+L=3+\sqrt{2}$ (irrational).
• $M + P=3\sqrt{2}+3$ (irrational).

Step3: Analyze products

• $L\cdot M=\sqrt{2}\times3\sqrt{2}=6$ (rational).
• $N\cdot P=4\times3 = 12$ (rational).
• $P\cdot L=3\times\sqrt{2}=3\sqrt{2}$ (irrational).
• $M\cdot P=3\sqrt{2}\times3 = 9\sqrt{2}$ (irrational).

Step4: For sequence $a_n=3 - 4(n - 1)$

First, expand $a_n=3-4n + 4=7-4n$.
The recursive form: $a_1=7-4\times1 = 3$, $a_{n}=a_{n - 1}-4$ for $n\gt1$.

Step5: For sequence $112,107,102,97,\cdots$

The common - difference $d=-5$.
Recursive form: $a_1 = 112$, $a_{n}=a_{n - 1}-5$ for $n\gt1$.
Explicit form: $a_n=a_1+(n - 1)d=112+(n - 1)(-5)=112-5n + 5=117-5n$.

Step6: For $f(x)=-3x + 2$ with $x$ a natural number

When $x = 1$, $a_1=-3\times1+2=-1$.
The recursive form: $a_1=-1$, $a_{n}=a_{n - 1}-3$ for $n\gt1$.

Step7: For system of equations (graph)

By substituting points into the equations of the lines (not shown but can be done conceptually):

• For point $(-2,-3)$: Substitute into the equations of the lines of the system.
• For point $(-1,2)$: Substitute into the equations of the lines of the system.
• For point $(2,1)$: Substitute into the equations of the lines of the system.
• For point $(-4,0)$: Substitute into the equations of the lines of the system.

Assuming by substitution, we find the solution points.

Answer:

1. Rational: $N + P$, $L\cdot M$, $N\cdot P$; Irrational: $L + M$, $P+L$, $M + P$, $P\cdot L$, $M\cdot P$
2. Recursive form: $a_1 = 3$, $a_{n}=a_{n - 1}-4$ for $n\gt1$
3. Recursive form: $a_1 = 112$, $a_{n}=a_{n - 1}-5$ for $n\gt1$; Explicit form: $a_n=117-5n$
4. $a_1=-1$; Recursive form: $a_1=-1$, $a_{n}=a_{n - 1}-3$ for $n\gt1$
5. (Solution depends on substitution into equations of lines of the system, no equations given so unable to give definite answer)