A Statement $$\mathrm { S } _ { \mathrm { n } }$$

question 161

Essay

A statement
$$\mathrm { S } _ { \mathrm { n } }$$ about the positive integers is given. Write statements
$\mathrm { S } _ { 1 } , \mathrm {~S} _ { 2 }$ and S
$\mathrm { S } _ { 3 }$ , and show that each of these
statements is true.
- $$S _ { n } : 1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

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Definitions:

Pulp Cavity

The central part of a tooth containing nerves and blood vessels, vital for the tooth's nutrition and sensitivity.

Apical Foramen

Opening at the apex of the root of a tooth; gives passage to the nerve and blood vessels.

Cementum

Layer of modified bone covering the dentin of the root and neck of a tooth; blends with the fibers of the periodontal membrane.

Mastication

Breaking down food by chewing it as a preparatory step for swallowing and digestion.