Use mathematical induction to prove that $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$. (b) 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 $n \geq 1$. (b) One end of a rope is attached to a truck and the oth... 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 this by induction, we start with the base case:

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

Inductive Step: Assume it holds for some integer $k \geq 1$ , i.e., assume: $2^k + (-1)^{k+1} \equiv 0 \mod 3.$

We need to show it holds for $k+1$ :

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

Using the inductive hypothesis, we know $2^k + (-1)^{k+1} \equiv 0 \mod 3$ . Thus: $2^{k+1} - 1 \equiv 2 \cdot 0 + 1 \equiv 1 \mod 3,$

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

Therefore, the statement holds for $k+1$ . Hence, by induction, $2^n + (-1)^{n+1}$ is divisible by 3 for all integers $n \geq 1$ .

Step 2

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

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Answer

From the right triangle formed by the truck, the small wheel, and the point where the rope is attached, we apply the Pythagorean theorem:

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

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

From the geometry of the triangle, $\cos\theta = \frac{x}{L},$ thus, $\frac{dL}{dx} = \cos\theta.$

Step 3

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

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Answer

We know from the previous step that $\frac{dL}{dx} = \cos\theta.$

Since the truck moves at a constant speed of 3 m/s, we can relate $\frac{dx}{dt}$ and find: $\frac{dL}{dt} = \frac{dL}{dx} \cdot \frac{dx}{dt} = \cos\theta \cdot 3 = 3\cos\theta.$

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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