A-Level Edexcel Maths Pure Geometric Sequences & Series: 10. (i) Prove that for all n ∈ ℕ

10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 1

Question 11

10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4. (ii) "Given x ∈ ℝ, the value of |3x - 28| is greater than or equal to the value of (x - 9)." State, ... show full transcript

Worked Solution & Example Answer:10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 1

Step 1

Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4.

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Answer

To prove that for all natural numbers n, n² + 2 is not divisible by 4, we can analyze the parity of n.

Step 1: Analyzing n

Let n be either even or odd.

If n is even, we can represent it as n = 2k for some integer k. Thus,

$n^2 + 2 = (2k)^2 + 2 = 4k^2 + 2$

This expression, $4k^2 + 2$ , is clearly not divisible by 4, since it leaves a remainder of 2.
If n is odd, we can express it as n = 2k + 1. Therefore,

$n^2 + 2 = (2k + 1)^2 + 2 = 4k^2 + 4k + 1 + 2 = 4k^2 + 4k + 3$

Here, $4k^2 + 4k + 3$ is also not divisible by 4, as it leaves a remainder of 3.

Conclusion

Since both cases of n (even and odd) lead to n² + 2 not being divisible by 4, we conclude that for all n ∈ ℕ, n² + 2 is not divisible by 4.

Step 2

Given x ∈ ℝ, the value of |3x - 28| is greater than or equal to the value of (x - 9). State, giving a reason, if the above statement is always true, sometimes true or never true.

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Answer

To investigate the statement about |3x - 28| and (x - 9), we consider the two cases based on the definition of the absolute value.

Step 1: Analyzing the Statement

We need to determine if:

$|3x - 28| ext{ is } \geq(x - 9).$

Case 1: When 3x - 28 is non-negative

Here,

$|3x - 28| = 3x - 28$

So, we evaluate the inequality:

$3x - 28 \geq (x - 9)$

This simplifies to:

$2x \geq 19 \rightarrow x \geq \frac{19}{2} = 9.5$

Case 2: When 3x - 28 is negative

In this case,

$|3x - 28| = -(3x - 28) = -3x + 28$

Evaluating this inequality leads to:

$-3x + 28 \geq (x - 9) \rightarrow 28 + 9 \geq 4x \rightarrow 37 \geq 4x \rightarrow x \leq \frac{37}{4} = 9.25$

Conclusion

Thus, the statement is only true for the range $9.25 \leq x \leq 9.5$ . Hence, it is sometimes true.

10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 1

Worked Solution & Example Answer:10. (i) Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4 - Edexcel - A-Level Maths Pure - Question 11 - 2019 - Paper 1

Prove that for all n ∈ ℕ, n² + 2 is not divisible by 4.

Step 1: Analyzing n

Conclusion

Given x ∈ ℝ, the value of |3x - 28| is greater than or equal to the value of (x - 9). State, giving a reason, if the above statement is always true, sometimes true or never true.

Step 1: Analyzing the Statement

Case 1: When 3x - 28 is non-negative

Case 2: When 3x - 28 is negative

Conclusion

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