Consider the Sequence Defined By $a _ { n } = \left( \frac { 2 } { 3 } \right) ^ { n }$

question 315

Essay

Consider the sequence defined by $a _ { n } = \left( \frac { 2 } { 3 } \right) ^ { n }$ . (n starts at 1)
(a) Write the first five terms of the sequence.(b) Determine the limit of the sequence.(c) Let $b _ { n } = \frac { a _ { n + 1 } } { a _ { n } }$ . Write the first five terms of this sequence.(d) Determine the limit of $b _ { n }$ .

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Definitions:

Negative Correlation

A correlation between two variables where the growth of one inversely affects the decline of the other.

Statistical Significance

A measure indicating that the likelihood of the observed results occurring by chance is very low, typically used in hypothesis testing to validate study findings.

Observed Differences

Variations noted in the outcomes or characteristics of subjects within a study or experiment.

Normal Curve

A bell-shaped curve that represents the distribution of a set of data where most occurrences take place in the middle, near the average.