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: $$$a_{37}=8+(37-1)\cdot 3=8+3\cdot 36 = 8+108=116$$$