Sum of Arithmetic Sequence/Examples/a0 = i, d = 2+2i

Example of Sum of Arithmetic Sequence

Let $A_n$ be the arithmetic sequence of $n$ terms defined as:

 $\ds A_n$ $=$ $\ds \sum_{k \mathop = 0}^{n - 1} \paren {a_0 + \paren {2 + 2 i} k}$ $\ds$ $=$ $\ds i + \paren {2 + 3 i} + \paren {4 + 5 i} + \paren {6 + 7 i} + \dotsb + \paren {2 n - 2 + \paren {2 n - 1} i}$

Then:

$A_n = n \paren {n - 1} + n^2 i$

Proof

 $\ds A_n$ $=$ $\ds n \paren {i + \frac {n - 1} 2 \paren {2 + 2 i} }$ Sum of Arithmetic Sequence: $a_0 = i$, $d = 2 + 2 i$ $\ds$ $=$ $\ds n \paren {i + \paren {n - 1} \paren {1 + i} }$ $\ds$ $=$ $\ds n \paren {i + n - 1 + n i - i}$ $\ds$ $=$ $\ds n \paren {n - 1 + n i}$ $\ds$ $=$ $\ds n \paren {n - 1} + n^2 i$

$\blacksquare$