A Statement Sn About the Positive Integers Is Given $$S _ { n } : 1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

question 269

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.
- $$S _ { n } : 1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

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Definitions:

Corticospinal Tract

A major neural pathway that runs from the cerebral cortex of the brain down to the spinal cord, primarily involved in motor control.

Tourette Syndrome

A neurological disorder characterized by repetitive, stereotyped, involuntary movements and vocalizations called tics.

Huntington Disease

A genetic neurodegenerative disorder characterized by the progressive breakdown of nerve cells in the brain, leading to physical and cognitive decline.

Hyperkinetic Symptoms

Manifestations characterized by excessive or abnormal involuntary movements, often found in disorders like Tourette's syndrome or Huntington's disease.