How one should teach using brackets in summation?
For example, why is it correct to write $\sum_i a_ib_i$ but $\sum_i a_i+b_i$ should be written as $\sum_i (a_i+b_i)$ But $\sum_i\frac{a_i+b_i}{2}$ is, in turn, correct case not to use brackets?
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Sign up to join this communityHow one should teach using brackets in summation?
For example, why is it correct to write $\sum_i a_ib_i$ but $\sum_i a_i+b_i$ should be written as $\sum_i (a_i+b_i)$ But $\sum_i\frac{a_i+b_i}{2}$ is, in turn, correct case not to use brackets?
I would say that in terms of order of operations, the summation symbol is between multiplication/division and addition/subtraction. So when you write $\sum_i a_i+b_i$, the implication is to do $\sum_i a_i$ first, and so you need parentheses. However in $\sum_i a_ib_i$ or $\sum_i \frac{a_i+b_i}{2}$ the multiplication and division by 2 come before the summation.
Note though that $\sum_i a_i+b_i$ implicitly means $\sum_i (a_i+b_i)$ because the dummy index on the $b_i$ doesn't make sense outside the context of the sum. The only real confusion comes in when you have something like $\sum_i a_i+3$.
Teacher: What do you want to add ?
Student: $a_i+b_i$
Teacher: Put it between brackets.
Student: $(a_i+b_i)$
Teacher: Sum it!
Student: $\sum_i (a_i + b_i)$
Same for the fraction:
Teacher: What do you want to add all together?
Student: $\frac{a_i+b_i}{2}$
Teacher: Put it between brackets.
Student: $(\frac{a_i+b_i}{2})$
Teacher: Sum it!
Student: $\sum_i (\frac{a_i+b_i}{2})$
=> this provides students with a fixed procedure, which is very clear, and does not generate any misunderstandings.
There are actually two questions that you're asking: how to teach "the summation symbol" and the followup question that asks for concrete explanations for examples.
I think the followup questions are non-questions if the summation symbol is carefully introduced and defined, I'd do something like