Which sigma expression correctly represents the sum 4 + 9 + 16?
AΣᵢ₌₁³ i² (evaluates to 1 + 4 + 9 = 14)
BΣᵢ₌₂⁴ i² (evaluates to 4 + 9 + 16 = 29)
CΣᵢ₌₀² i² (evaluates to 0 + 1 + 4 = 5)
DΣᵢ₌₂³ i² (evaluates to 4 + 9 = 13)
4 + 9 + 16 = 2² + 3² + 4², so the general term is i² and the index runs from 2 to 4: Σᵢ₌₂⁴ i². Option A is the most tempting mistake — starting at i = 1 shifts everything by one and includes 1² instead of 4². Off-by-one errors in the lower bound are the most common mistake in translating from a written sum to sigma notation.
Question 2 Multiple Choice
A student writes Σᵢ₌₁ⁿ i² to represent a sum, then rewrites it as Σₖ₌₁ⁿ k². Her classmate says the sum has changed because k is not the same variable as i. Who is correct?
AThe classmate — i and k are different variables and will produce different values
BThe student — the index variable is a dummy variable and renaming it does not change the sum
CBoth are partially right — the symbolic expression differs, but the numerical value only changes for some values of n
DNeither — sigma notation requires using i as the index by convention
The index variable in sigma notation is called a 'dummy variable' because it exists only within the sum and its name is irrelevant. Σᵢ₌₁ⁿ i², Σₖ₌₁ⁿ k², and Σⱼ₌₁ⁿ j² are completely identical expressions. This same concept appears later in calculus, where the variable of integration is also a dummy variable. The bounds and the expression defining each term are what matter — the letter used for the index does not.
Question 3 True / False
In sigma notation, the index of summation is expected to start at 1.
TTrue
FFalse
Answer: False
False. The lower bound of a sigma expression can be any integer — 0, 2, -3, or anything else. For example, Σᵢ₌₀ⁿ xⁱ/i! starts at 0 (it's the Taylor series for eˣ), and many combinatorial sums start at 2 or higher. Assuming the index must start at 1 is a common source of off-by-one errors.
Question 4 True / False
The expressions Σᵢ₌₁ⁿ i² and Σₖ₌₁ⁿ k² represent the same numerical value for any positive integer n.
TTrue
FFalse
Answer: True
True. Both expressions expand to 1² + 2² + ... + n² — the letter used for the index makes no difference whatsoever. Renaming a dummy variable is not a mathematical change; it is like renaming a local variable in a function without changing what the function computes.
Question 5 Short Answer
Why is the index variable in sigma notation called a 'dummy variable,' and how does this differ from the role of the bounds?
Think about your answer, then reveal below.
Model answer: The index variable is called a dummy variable because it exists only within the summation and its specific name carries no meaning — renaming i to j or k produces an identical sum. The bounds, by contrast, are not interchangeable: they specify exactly which terms are included. Changing the lower bound from 1 to 2 changes the first term included in the sum; changing the upper bound from n to n+1 adds one more term. The dummy variable name is arbitrary scaffolding; the bounds are the actual specification of which terms to add.
This distinction matters because students sometimes think they can 'adjust' a sum by renaming the index, when only the bounds and the expression define the sum's value. The concept of dummy variables reappears in integration (∫f(x)dx = ∫f(t)dt) and in programming (a loop variable has no meaning outside the loop body).