A Statement Sn About the Positive Integers Is Given $$S _ { n } : 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$

question 159

Essay

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these
statements is true.
- $$S _ { n } : 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$

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Definitions:

T-statistic

A type of statistic used in hypothesis testing when the sample size is small and the population variance is unknown, it is based on the Student's t-distribution.

Z-score

A statistic that tells you how many standard deviations a given value is from the mean.

Sample Mean

The average value of a sample set, calculated by summing all observations and dividing by the number of observations.

Sampling Distribution

The probability distribution of a statistic based on a large number of samples drawn from a specific population.