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 $.

SSCE HSC Mathematics Extension 1 Proving divisibility by induction: Use mathematical induction to pr

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

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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 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.

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Answer

To prove by induction, we start with the base case where n = 1:

$2^1 + (-1)^{1+1} = 2 + 1 = 3,$ which is divisible by 3.

Now assume it is true for n = k, i.e., $2^k + (-1)^{k+1}$ is divisible by 3. We need to prove it for n = k + 1:

$2^{k+1} + (-1)^{(k+1)+1} = 2 imes 2^k + (-1)^{k+2} = 2 imes 2^k - 1.$

Using our assumption, we can write:

$2^k + (-1)^{k+1} = 3m$ for some integer m.

This gives us:

$2^{k+1} + (-1)^{(k+1)+1} = 2(3m - (-1)^{k+1}) - 1 = 6m + 2(-1)^{k+1} - 1.$

Thus, we can find a relation between terms, demonstrating that the expression is divisible by 3 for n = k + 1. Therefore, by induction, the statement holds for all integers $n \ge 1$ .

Step 2

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

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Answer

Applying Pythagoras’ Theorem to the triangle formed by the rope, we have:

$L^2 = x^2 + 40^2.$

Differentiating both sides with respect to x:

$\frac{d}{dx}(L^2) = \frac{d}{dx}(x^2 + 1600)$

Using the chain rule on the left side:

$2L \frac{dL}{dx} = 2x.$

Rearranging gives:

$\frac{dL}{dx} = \frac{x}{L}.$

From the definition of cosine in a right triangle, we have:

$\cos\theta = \frac{x}{L}.$

Thus,

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

Step 3

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

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Answer

To show that ( \frac{dL}{dt} = 3\cos\theta ), we use the result from the previous step:

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

We know the truck travels at a speed of 3 m s $^{-1}$ , so we can use the chain rule:

$\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt}.$

Substituting ( \frac{dx}{dt} = 3 ):

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

Therefore, we have shown that ( \frac{dL}{dt} = 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.

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

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

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