General term of an arithmetical progression

To find the general term of an arithmetical progression we consider the formula that defines these progressions: $a_{n+1}-a_n=d$.

This equality expresses that, in the arithmetical progressions, every term is obtained by adding the difference to the previous term. This way, we can define the progression in a recursive way and then: $$a_{n+1}=a_n+d$$

If we apply this law recursively to construct the succession, we obtain: $$a_2=a_1+d$$ $$a_3=a_2+d=(a_1+d)+d=a_1+2d$$ $$a_4=a_3+d=(a_1+2d)+d=a_1+3d$$ $$a_5=a_4+d=(a_1+3d)+d=a_1+4d$$ $$\ldots$$

And, in general, we have $$a_n=a_1+(n-1)d$$ This expression relates any term of the succession to the first using the difference of the progression.

We want to find the number that is in position $37$ of the succession $$(8,11,14,17,20,\ldots)$$ We notice that it is an arithmetical progression because the difference between all the terms is constant and equal to $3$.

As the first term is $a_1=8$, and the difference is $d=3$, we have: $$a_n=8+(n-1)\cdot 3$$ Since we want to find the term $a_{37}$, we can proceed: $$a_{37}=8+(37-1)\cdot 3=8+3\cdot 36 = 8+108=116$$