Summation is Linear/Sum of Summations

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Theorem

Let $\tuple {x_1, \ldots, x_n}$ and $\tuple {y_1, \ldots, y_n}$ be finite sequences of numbers of equal length.

Let $\lambda$ be a number.

Then:

$\displaystyle \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$


Proof

The proof proceeds by mathematical induction.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:

$\displaystyle \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$


Basis for the Induction

$\map P 1$ is the case:

\(\displaystyle \sum_{i \mathop = 1}^1 x_i + \sum_{i \mathop = 1}^1 y_i\) \(=\) \(\displaystyle x_1 + y_1\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^1 \paren {x_i + y_i}\)

This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is the induction hypothesis:

$\displaystyle \sum_{i \mathop = 1}^k x_i + \sum_{i \mathop = 1}^k y_i = \sum_{i \mathop = 1}^k \paren {x_i + y_i}$


from which it is to be shown that:

$\displaystyle \sum_{i \mathop = 1}^{k + 1} x_i + \sum_{i \mathop = 1}^{k + 1} y_i = \sum_{i \mathop = 1}^{k + 1} \paren {x_i + y_i}$


Induction Step

This is the induction step:

\(\displaystyle \sum_{i \mathop = 1}^{k + 1} x_i + \sum_{i \mathop = 1}^{k + 1} y_i\) \(=\) \(\displaystyle \paren {\sum_{i \mathop = 1}^k x_i + x_{k + 1} } + \paren {\sum_{i \mathop = 1}^k y_i + y_{k + 1} }\) Definition of Summation
\(\displaystyle \) \(=\) \(\displaystyle \left({\sum_{i \mathop = 1}^k x_i + \sum_{i \mathop = 1}^k y_i}\right) + \paren {x_{k + 1} + y_{k + 1} }\) Commutative Law of Addition and Associative
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^k \paren {x_i + y_i} + \paren {x_{k + 1} + y_{k + 1} }\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^{k + 1} \paren {x_i + y_i}\) Definition of Summation

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\displaystyle \forall n \in \N_{> 0}: \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$

$\blacksquare$