Problems from General term of an arithmetical progression

Development:

Let's see what the difference is $$$d=(1+\sqrt{2})-(1-\sqrt{2})=\sqrt{2}+\sqrt{2}=2\sqrt{2}$$$ And, as the first term is $$a_1=1-\sqrt{2}$$, we know that:

$$$a_n=(1-\sqrt{2})+(n-1)\cdot 2 \cdot \sqrt{2}$$$

Arranging this expression we have:

$$$a_n=(1-\sqrt{2})+2\sqrt{2}(n-1)=1-\sqrt{2}-2\sqrt{2}+2\sqrt{2}\cdot n=2\sqrt{2}n+1-3\sqrt{2}$$$

Solution:

$$a_n=2\sqrt{2}n+1-3\sqrt{2}$$

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Development:

The difference is $$d=\dfrac{3}{4}-\dfrac{1}{2}=\dfrac{1}{4}$$, and as $$a_1=\dfrac{1}{2}$$, we have that $$$a_n=\dfrac{1}{2}+\dfrac{1}{4}(n-1)=\dfrac{n}{4}+\dfrac{1}{4}=\dfrac{n+1}{4}$$$

So:

$$$a_4=\dfrac{4+1}{4}=\dfrac{5}{4}, \ a_8=\dfrac{9}{4}$$$ and $$$a_{113}=\dfrac{114}{4}=\dfrac{57}{2}=28+\dfrac{1}{2}$$$

Solution:

$$a_4=\dfrac{5}{4}, \ a_8=\dfrac{9}{4}$$ and $$a_{113}=\dfrac{57}{2}$$

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