Question

Foe the AP given by a1,a2,......,an,...with non zero common difference, the equation satisfied are

Answer 1

The given sequence is
a₁, a₂, ..., $a_{n}$

Because the given sequence is an arithmetic progression (AP), the equation satisfied is
$a_{n}=a_{1}+(n-1)d$
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
$S_{n}= \frac{n}{2} (a_{1}+a_{n})$

Example:
For the arithmetic sequence
1,3,5, ..., 
the common difference is d= 3 - 1 = 2.
The n-th term is
$a_{n}=1+2(n-1)$

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