Sum of Arithmetic Sequence/Examples/a0 = i, d = 2+2i
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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: $a_0 = i$, $d = 2 + 2 i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {i + \paren {n - 1} \paren {1 + i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {i + n - 1 + n i - i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {n - 1 + n i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {n - 1} + n^2 i\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $3$