How do I use these two formulae? They have something to do with line segments. Thanks
If i am not mistaken those formulas are used for summation of arithmetic progression.
initial value or first term = a
common difference (difference between one term and the next term) = d
last term = $\displaystyle t_n$
number of terms = n
The first formula is used when you know or given the initial value of the progression, the common difference, and the number of terms in that particular arithmetic progression.
The second on is used when you know or given the initial and final value of the arithmetic progression, and the number of terms in that particular arithmetic progression.
For example;
1,2,3,4,5,6,7,8,9,10
in this case we can use both the formula since we know the initial value, the final value, the number of terms and the common difference.
n=10, a = 1, $\displaystyle t_n$ = 10, d = 1
For the first formula:
$\displaystyle
S_n = \frac n2 \left[ 2a+(n-1)d) \right]
$
$\displaystyle
S_n = \frac {10}{2} \left[ 2(1)+(10-1)1) \right]
$
$\displaystyle
S_n = 55
$
For the second formula:
$\displaystyle
S_n = \frac {n}{2} \left[ a+t_n \right]
$
$\displaystyle
S_n = \frac {10}{2} \left[ 1+10 \right]
$
$\displaystyle
S_n = 55
$
Mathwords: Arithmetic Series