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23 (a) n is an integer - OCR - GCSE Maths - Question 23 - 2017 - Paper 1

Question 23

23 (a) n is an integer. (i) Explain why 2n + 1 is an odd number: (ii) Write down an algebraic expression for the next odd number after 2n + 1. (b) Use algebra to ... show full transcript

Worked Solution & Example Answer:23 (a) n is an integer - OCR - GCSE Maths - Question 23 - 2017 - Paper 1

Step 1

Explain why 2n + 1 is an odd number:

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Answer

To determine why the expression $2n + 1$ is an odd number, we start by analyzing the term $2n$ . Since $n$ is an integer, $2n$ will always yield an even integer because it is the product of 2 and another integer. An even integer can be expressed in the general form $2k$ , where $k$ is an integer.

When we add 1 to an even number, we shift it to the next integer, transforming it into an odd number. Therefore, since $2n$ is even, the expression $2n + 1$ must be odd.

Step 2

Write down an algebraic expression for the next odd number after 2n + 1:

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Answer

The next odd number following the number $2n + 1$ can be expressed as $2n + 3$ . This is derived by simply adding 2 to $2n + 1$ , thereby yielding the next odd integer.

Step 3

Use algebra to show that the sum of two consecutive odd numbers will always be a multiple of 4:

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Answer

Let the first odd number be represented as $x$ . Therefore, the next consecutive odd number can be expressed as $x + 2$ . The sum of these two consecutive odd numbers is given by:

$S = x + (x + 2) = 2x + 2$

We can factor out a 2 from the expression:

$S = 2(x + 1)$

Since $x$ is an odd number, $(x + 1)$ is an even integer. Therefore, $2(x + 1)$ will always yield a result that is an even multiple of 2, or more specifically, a multiple of 4.

23 (a) n is an integer - OCR - GCSE Maths - Question 23 - 2017 - Paper 1

Worked Solution & Example Answer:23 (a) n is an integer - OCR - GCSE Maths - Question 23 - 2017 - Paper 1

Explain why 2n + 1 is an odd number:

Write down an algebraic expression for the next odd number after 2n + 1:

Use algebra to show that the sum of two consecutive odd numbers will always be a multiple of 4:

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