Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$. One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of 40 m above the point where the rope is attached to the truck. The distance from the truck to the small wheel is $L$ m, and the horizontal distance between them is $m$ m. The rope makes an angle $\theta$ with the horizontal at the point where it is attached to the truck. The truck moves to the right at a constant speed of 3 m s$^{-1}$, as shown in the diagram. (i) Using Pythagoras' Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$. (ii) Show that $\frac{dL}{dt} = 3 \cos \theta$.

Photo AI

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Question 13

$Use-mathematical-induction-to-prove-that-$2^n-+-(-1)^{n+1}$-is-divisible-by-3-for-all-integers-$n-\geq-1$-HSC-SSCE Mathematics Extension 1-Question 13-2014-Paper 1.png$

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$. One end of a rope is attached to a truck and the other en... show full transcript

Worked Solution & Example Answer:Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Step 1

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$

96%

114 rated

Answer

To prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ , we can employ mathematical induction.

Base Case: For $n = 1$ , we calculate: $2^1 + (-1)^{1+1} = 2 + 1 = 3,$ which is clearly divisible by 3.

Inductive Step: Assume the statement holds for some integer $k$ , i.e., that $2^k + (-1)^{k+1}$ is divisible by 3. We must show it is also true for $k + 1$ : $2^{k+1} + (-1)^{(k+1)+1} = 2 \cdot 2^k + (-1)^{k+2} = 2(2^k) - 1.$

From the inductive hypothesis, we know that $2^k + (-1)^{k+1} = 3m$ for some integer $m$ . Therefore, substituting gives us: $2(2^k + (-1)^{k+1}) - 1 = 2(3m) - 1 = 6m - 1.$

We need to show that $6m - 1$ is also divisible by 3. Note that: $6m - 1 = 3(2m) - 1,$ which is not directly useful for divisibility by 3. To finalize: $2^{k+1} + (-1)^{k+2} = 2(3m) - 1 = 6m - 1 = 3(2m-1).$ Thus, this expression is divisible by 3. Hence, by the principle of mathematical induction, the statement is proven for all integers $n \geq 1$ .

Step 2

Using Pythagoras' Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$

99%

104 rated

Answer

Consider the right triangle formed by the vertical distance (40 m), the horizontal distance ( $x$ m), and the distance to the wheel ( $L$ m). Using Pythagorean Theorem, we have: $L^2 = x^2 + 40^2.$

Differentiating both sides with respect to $x$ , we get: $2L \frac{dL}{dx} = 2x,$ which simplifies to: $\frac{dL}{dx} = \frac{x}{L}.$

Also, from the definition of cosine in the triangle, we have: $\cos \theta = \frac{x}{L}.$

Thus: $$\frac{dL}{dx} = \cos \theta.$

Step 3

Show that $\frac{dL}{dt} = 3 \cos \theta$

96%

101 rated

Answer

Using the chain rule, we can relate $\frac{dL}{dt}$ and $\frac{dL}{dx}$ : $\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt}.$

From the previous part, we have: $\frac{dL}{dx} = \cos \theta,$ and since the truck is moving to the right at a constant speed of 3 m/s, we know: $\frac{dx}{dt} = 3.$

Substituting these into the equation gives us: $\frac{dL}{dt} = \cos \theta \cdot 3 = 3 \cos \theta.$

Hence, this demonstrates the required relationship.

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Worked Solution & Example Answer:Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ - HSC - SSCE Mathematics Extension 1 - Question 13 - 2014 - Paper 1

Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$

Using Pythagoras' Theorem, or otherwise, show that $\frac{dL}{dx} = \cos \theta$

Show that $\frac{dL}{dt} = 3 \cos \theta$

Join the SSCE students using SimpleStudy...