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 $x$ 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.

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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

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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 e... 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 $n \geq 1$

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Answer

To prove the statement using induction, we first verify the base case (n=1):
$2^1 + (-1)^{1+1} = 2 + 1 = 3$
which is divisible by 3.

Next, assume that the statement holds for some integer $k \geq 1$ :
$2^k + (-1)^{k+1} \text{ is divisible by } 3.$
We need to show that it holds for $k + 1$ :
$2^{k+1} + (-1)^{(k+1)+1} = 2 \times 2^k + (-1)^{k+2} = 2 \times 2^k - 1.$
Adding the induction hypothesis:
$2 \times 2^k - 1 + (2^k + (-1)^{k+1}) = (2 + 1) \times 2^k - 1 = 3 \times 2^k - 1.$
We know $3 \times 2^k$ is divisible by 3, and $-1 + 3 \times 2^k$ ensures that the entire expression is divisible by 3, making the statement valid for k + 1. Thus, the induction step holds and the proof is complete.

Step 2

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

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Answer

Using the Pythagorean theorem, we have:
$L^2 = x^2 + 40^2.$
Differentiating both sides with respect to $x$ :
$2L \frac{dL}{dx} = 2x,$
which simplifies to:
$\frac{dL}{dx} = \frac{x}{L}.$
Since $\cos\theta = \frac{x}{L}$ , we can conclude that:
$\frac{dL}{dx} = \cos\theta.$

Step 3

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

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Answer

Applying the chain rule:
$\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt}.$
From the prior result, we have $\frac{dL}{dx} = \cos\theta$ and since the truck moves right at a speed of 3 m/s, then:
$\frac{dx}{dt} = 3.$
Combining these results yields:
$\frac{dL}{dt} = \cos\theta \cdot 3 = 3 \cos\theta.$

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 $\frac{dL}{dt} = 3 \cos\theta$.

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