Question

What is the 100th term if the sequence? 5 ,1 ,-3,-7

Answer 1

5 ,1 ,-3,-7  ?  what the dickens is going on?

notice, from 5 to 1, is really -4 units,

from 1 to -3, is -4 units as well, 1 -4 = -3  and so on.

so, to get the next term's value, you simply subtract 4 from the current one, namely -4 is the "common difference".

and we also know the first term is 5, ok... then

$\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=-4\\ a_1=5\\ n=100 \end{cases} \\\\\\ a_{100}=5+(100-1)(-4)\implies a_{100}=5+(99)(-4) \\\\\\ a_{100}=5-396\implies a_{100}=-391$