Use Mathematical Induction to Prove the Statement Is True for All

question 30

Essay

Use mathematical induction to prove the statement is true for all positive integers n.
- $$5 \cdot 7 + 6 \cdot 8 + 7 \cdot 9 + \ldots + 5 n ( 5 n + 2 ) = \frac { 5 n ( 5 n + 1 ) ( 10 n + 7 ) } { 6 }$$

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Definitions:

Standard Normal

Standard normal refers to a normal distribution with a mean of 0 and a standard deviation of 1.

Area

The quantifiable space within the boundaries of a two-dimensional shape, measured in square units.

Standard Normal

A normal distribution with a mean of 0 and a standard deviation of 1, used as a basis for comparison with other normal distributions.

Standard Deviation

A measure of the amount of variation or dispersion of a set of values; a low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.