Summation Rules: What Happens When k=0?

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Discussion Overview

The discussion revolves around the rules of summation, particularly focusing on the implications of starting the summation index at zero and how it affects the total sum. Participants explore the mathematical formulation and interpretation of summation notation.

Discussion Character

  • Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant questions the meaning of the summation when the index starts at k=0.
  • Another participant asserts that the sum results in ##3(n+1)## when k=0, suggesting that the sum includes an additional term.
  • A further contribution indicates that starting the index at k=-1 leads to a sum of ##3(n+2)##, implying that the lower bound affects the total count of terms.
  • Some participants summarize the general rule for summation as $$\sum_{k=a}^{b} c = c (b-a+1)$$ for a constant ##c##, using an analogy of counting fence posts.
  • There is a repeated emphasis on the interpretation of the lower bound and its relevance to the total number of terms in the summation.

Areas of Agreement / Disagreement

Participants appear to agree on the general rule of summation presented, but there is some uncertainty regarding the implications of starting the index at different values, particularly k=0 and k=-1. The discussion does not reach a consensus on the interpretation of these cases.

Contextual Notes

The discussion includes assumptions about the interpretation of summation indices and their impact on the total count, which may depend on the context of the problem being addressed.

StevenJacobs990
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n
∑ 3
k=0

How does this make sense when k=0?
 
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The sum is ##3+3+...=3(n+1)##
 
Gene Naden said:
The sum is ##3+3+...=3(n+1)##
Oh okay. The lower bound is the index origin and doesn't matter if it is negative?
n
∑ 3
k=-1
3+3+...=3(n+2)
 
Correct!
 
To summarise,
$$
\sum_{k=a}^{b} c = c (b-a+1)
$$
for constant ##c##.
 
DrClaude said:
To summarise,
$$
\sum_{k=a}^{b} c = c (b-a+1)
$$
for constant ##c##.
i.e. one is counting fence posts, not sections of wire.
 

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