Mathematics

Mathematics

Problem

Consider the idea of a geometric progression (GP). a. (a) Define a geometric progression and write the symbolic expression for its general (nth) term. b. (b) A GP has a positive common ratio greater than 1. Explain, without actual numerical calculation, why the sequence of its terms is strictly increasing. c. (c) A GP has a negative common ratio. Describe the pattern of signs (positive or negative) that will appear in its terms and justify your answer with reasons based on the value of the common ratio.
Intuitive explanation

A geometric progression is a sequence where each term is found by multiplying the previous one by a fixed constant called the common ratio rr. The key insight is that this multiplicative structure creates exponential growth or decay—so instead of adding the same amount each time (like arithmetic sequences), you're scaling by the same factor. Understanding rr tells you everything: if ∣r∣>1|r| > 1 the sequence explodes, if ∣r∣<1|r| < 1 it shrinks, and if rr is negative the terms alternate signs.

Intuitive explanation

A geometric progression is a sequence where each term is found by multiplying the previous one by a fixed constant called the common ratio rr. The key insight is that this multiplicative structure creates exponential growth or decay—so instead of adding the same amount each time (like arithmetic sequences), you're scaling by the same factor. Understanding rr tells you everything: if ∣r∣>1|r| > 1 the sequence explodes, if ∣r∣<1|r| < 1 it shrinks, and if rr is negative the terms alternate signs.

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