Algebra 2 Equation of growth and decay, Equation of arithmetic sequence 
*

*Why do I need a $1$ in this equation of growth and decay? $A=P(1\pm r)^t$ 

*Why must I divide by $2$ to find the sum of an arithmetic sequence?
$S(n) = \frac{n}{2}(a_1 + a_n)$ 

1 - You don't really need the 1 in the exponential formula. any exponential function can be written in the form 
$$f(x)=a(b^x) $$
where $b$ is the common ratio
It is good to understand that ...


*

*if $b=1$ there is no growth or decay

*if $b>1$ then $|f|$ is increasing by a growth factor of $(b-1)$

*if $b<1$ then $|f|$ is decreasing by a decay factor of $(1-b)$
2 - The arithmetic sum is based on the observation that if ..
$$t_n = a + (n-1)d $$
then for every $k$ satisfying $1\le k\le n$
$$ t_k+t_{n+1-k} = 2a + [(k-1) + (n+1-k-1)]d
\\= 2a+(n-1)d = t_1+t_n$$
which is independent of $k$ so we can conclude that the average value of each term in the series in $\frac{t_1+t_n}2$.
Since the series contains $n$ terms, the sum will be ...
$$S_n = n\left(  \frac{t_1+t_n}2 \right )$$
