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Prove some properties of finite sums

by RoRi
  1. Prove the additive property of finite sums.
  2. Prove the homogeneous property of finite sums.
  3. Prove the telescoping property of finite sums.
(Note: Before we start these, I’ll note that these are all finite sums, so we are free to manipulate them in ways we can’t manipulate infinite sums. Which is to say, we can freely reorder terms in any way we like, or basically use our usual field axioms to put things in the form we want. If the sums are infinite (see chapters 10 and 11… or a course in real analysis) then we don’t have such freedom.)

  1. Proof. We expand, rearrange, and then write back in summation notation,

        \begin{align*} \sum_{k=1}^n (a_k + b_k) &= (a_1 + b_1) + (a_2 + b_2) + \cdots + (a_n + b_n) \\ &= (a_1 + a_2 + \cdots + a_n) + (b_1 + b_2 + \cdots + b_n) & (\text{Assoc. and Comm.}) \\ &= \sum_{k=1}^n a_k + \sum_{k=1}^n b_k. \qquad \blacksquare \end{align*}

  2. Proof. We compute using the distributive law,

        \[ \sum_{k=1}^n (ca_k) = ca_1 + ca_2 + \cdots + ca_n = c(a_1 + a_2 + \cdots + a_n) = c \sum_{k=1}^n a_k. \qquad \blacksquare \]

  3. Proof.

        \begin{align*} \sum_{k=1}^n (a_k - a_{k-1}) &= \left(\sum_{k=1}^n a_k\right) + (-1)\cdot \left( \sum_{k=1}^n a_{k-1}\right) & (\text{Parts (a) and (b)})\\ &= a_n + \sum_{k=1}^{n-1} a_k - \sum_{k=0}^{n-1} a_k & (\text{Reindexing the second sum}) \\ &= a_n + \sum_{k=1}^{n-1} a_k - a_0 - \sum_{k=1}^{n-1} a_k & (\text{Pulling out } a_0)\\ &= a_n - a_0. \qquad \blacksquare \end{align*}

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