Use an+1 / an to show that the given sequence {an} is strictly increasing or strictly decreasing

step 1 Suppose we have a sequence defined by the following where N starts at 1 and goes to positive inf

Use an+1 / an to show that the given sequence {an} is...

Key Concepts

Properties of Exponential Functions

Many sequences involve terms expressed as exponential functions, and understanding how to manipulate these terms is crucial. Simplifying expressions using the rules of exponents—such as a^(n+1) divided by a?, which simplifies to a—allows for clearer analysis of growth or decay behavior, which is essential when determining the monotonicity of the sequence.

Sequence Monotonicity

Monotonicity refers to the property of a sequence where its terms are either exclusively non-decreasing or non-increasing. A strictly monotonic sequence is one that is either strictly increasing (each term is greater than its predecessor) or strictly decreasing (each term is less than its predecessor). Establishing monotonicity helps in understanding convergence and other properties of the sequence.

Consecutive Term Ratio

This concept involves comparing the ratio of successive terms in a sequence, namely a???/a?. By calculating this ratio, one can analyze the behavior of the sequence without necessarily finding its limit or closed form. When the ratio is consistently greater than 1, the sequence is strictly increasing; if it is consistently less than 1, the sequence is strictly decreasing.

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