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

step 1 Suppose I have a sequence defined by the following, where inserts at 1 to positive infinity. Oka

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

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Key Concepts

Algebraic Simplification and Inequality Analysis

Manipulating and simplifying expressions, especially when they involve ratios of exponential terms, is crucial for determining the monotonicity of a sequence. By simplifying the expression for a???/a? and comparing it to one, one can directly assess whether the sequence increases or decreases for all n.

Exponential Growth in Sequences

Exponential functions, such as those containing 2?, display rapid growth as the variable n increases. When these types of terms are present in a sequence, they often dominate the behavior of the sequence and are key to understanding its overall growth pattern.

Monotonic Sequences

A sequence is said to be monotonic if its terms consistently increase or decrease. To determine if a sequence is strictly increasing or strictly decreasing, one typically examines the relationship between consecutive terms, ensuring that each term is greater than or less than the one preceding it.

Ratio Technique for Monotonicity

One effective method for determining the monotonicity of a sequence is by analyzing the ratio a???/a?. If this ratio is consistently greater than one, it indicates that every successive term is larger than the previous term, hence the sequence is strictly increasing. Conversely, if the ratio is less than one for all n, the sequence is strictly decreasing.

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