(4 marks)

**M1** | For the key step attempting to find $n^2 + 1$ when $n = 2k$ or $n = 2k \pm 1$

**A1** | Achieves $4k^2 + 1$ for $n = 2k$ or $4(k^2 + k) + 2$ for $n = 2k + 1$ or $4(k^2 - k) + 2$ for $n = 2k - 1$

**dM1** | Attempts find $n^2 + 1$ when $n = 2k$ and $n = 2k \pm 1$

**A1** | Correct work and states "is not divisible by 4" with a final conclusion e.g. so true for all $n \in \mathbb{N}$. There should be no errors in the algebra but allow e.g. invisible brackets if they are recovered.

**Alternative method:**

**M1** | Sets up an algebraic statement in terms of a variable (integer) k or any other variable aside n that engages with divisibility by 4 in some way and can lead to a contradiction. No need for explicit statement of assumption-accept if just a suitable equation is set up. In this case supposing divisibility by 4 by stating $n^2 + 1 = 4k$

**A1** | Reaches $n^2 = 4k - 1 \Rightarrow n^2 = 2(2k - 1) + 1$. Example: $n^2 = 4k - 1 = 4k - 2 + 1 = 2(2k - 1) + 1$

**dM1** | For a complete argument that leads to a contradiction. Accept explanations such as "as $n^2$ is even then n is even hence $n^2$ is a multiple of 4 so $n^2 + 1$ cannot be a multiple of 4"

Example: as $n^2$ is even so $n = 2m$ is even hence $n^2$ is a multiple of 4. As $n^2$ is a multiple of 4 then $n^2 + 1 = 4m^2 + 1 = 4m^2 + 2 - 1 = 2(2m^2 + 1) - 1$ cannot be a multiple of 4 so there is a contradiction.

**A1** | Draws the contradiction to their initial assumption and concludes the statement is true for all n. There must have been a clear assumption at the start that is contradicted, and all working must have been correct. Example: So the original assumption has been shown false. Hence "$n^2 + 1$ is never divisible by 4" is true for all n

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