The given sequence is
a₁, a₂, ..., [tex] a_{n} [/tex]
Because the given sequence is an arithmetic progression (AP), the equation satisfied is
[tex] a_{n}=a_{1}+(n-1)d [/tex]
where
d = the common difference.
The common difference may be determined as
d = a₂ - a₁
The common difference is the difference between successive terms, therefore
d = a₃ - a₂ = a₄ - a₃, and so on..
The sum of the first n terms is
[tex]S_{n}= \frac{n}{2} (a_{1}+a_{n})[/tex]
Example:
For the arithmetic sequence
1,3,5, ...,
the common difference is d= 3 - 1 = 2.
The n-th term is
[tex]a_{n}=1+2(n-1)[/tex]
For example, the 10-term is
a₁₀ = 1 + (10-1)*2 = 19
Th sum of th first 10 terms is
S₁₀ = (10/2)*(1 + 19) = 100
