Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.
Select arithmetic or geometric structure from its invariant
Problem
Payments increase by \(\$25~\text{each}~\text{month}\). Classify the pattern as arithmetic or geometric, state its invariant with units, and write the corresponding term model.
Big Picture
What this problem is really about
Classify a pattern by what stays constant, not merely by whether it increases. Repeated addition gives a constant difference and an arithmetic model; repeated multiplication gives a constant ratio and a geometric model. Units help distinguish them: a dollar change per month is additive, whereas a percent change becomes a unitless multiplier.
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