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

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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 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 this statement by mathematical induction, we first check the base case for $n = 1$ :

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

Inductive Step: Assume the statement is true for some integer $k \geq 1$ , i.e.,

$2^k + (-1)^{k+1} = 3m$ for some integer $m$ . We need to show that it holds for $k + 1$ :

$2^{k+1} + (-1)^{(k+1)+1} = 2 \cdot 2^k + (-1)^{k+2} = 2 \cdot 2^k - 1$ From the induction hypothesis, we substitute:

$= 2(3m - (-1)^{k+1}) - 1 = 6m - 2(-1)^{k+1} - 1$ Calculate this expression:

$= 6m - 1 - 2(-1)^{k+1} = 6m - 1 + 2(-1)^{k+2}$ Thus,

$= 3(2m - 1)$ This implies that $2(m - 1) + (-1)^{k+2}$ is divisible by 3. Hence, the statement holds for $n = k + 1$ . By induction, the statement is true 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

By applying Pythagoras' theorem to the right triangle formed, where the vertical side is 40 m, the horizontal side is $x$ , and the hypotenuse is $L$ , we have:

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

Differentiating both sides with respect to $x$ , we apply implicit differentiation:

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

From this, we isolate $rac{dL}{dx}$ :

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

From trigonometric relationships, we know:

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

Thus, we conclude that:

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

Step 3

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

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Answer

Using the chain rule, we can express the rate of change of $L$ with respect to time $t$ as follows:

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

From the previous result, we know:

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

The truck moves at a constant speed of 3 m/s, so:

$\frac{dx}{dt} = 3$

Substituting these values, we find:

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

Thus, 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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