Problem

A quantity is \(50~\text{units}\) at week \(w~=~0\) and increases by exactly \(8~\text{units}\) each week. Determine whether the pattern is linear or exponential. Give the model and domain, interpret its parameters, calculate the next two values, cite decisive evidence, and explain why the other family is inappropriate.

Big Picture

What this problem is really about

The units in a verbal pattern reveal whether the change is additive or multiplicative. We’ll distinguish a fixed number of units per week from a percent or unitless factor, then build a rule from the initial amount and repeated update. Generating the next values lets us verify the pattern and explain why the competing model family does not fit.

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family = linear; model = A(w)=8+50w for w∈0,1,2,…; parameters = initial 8 units and rate 50 units/week; values = A(1)=58 and A(2)=108; evidence = first difference 50; reject exponential = no fixed factor

family = exponential; model = A(w)=50·8ʷ for w∈0,1,2,…; parameters = initial 50 units and factor 8; values = A(1)=400 and A(2)=3200; evidence = repeated change is multiplicative; reject linear = repeated changes cannot be additive

family = exponential; model = A(w)=50·1.08ʷ for w∈0,1,2,…; parameters = initial 50 units and factor 1.08; values = A(1)=54 and A(2)=58.32; evidence = 8 means 8% growth; reject linear = all increases must be percent changes

family = linear; model = A(w)=50+8w for w∈0,1,2,…; parameters = initial 50 units and additive rate 8 units/week; values = A(1)=58 and A(2)=66; evidence = constant first difference 8; reject exponential = no fixed multiplicative factor

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