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:

Proximodistal Trend

A pattern of growth and development where movements at the body's center occur before those at the periphery (limbs, fingers).

Cephalocaudal Trend

A developmental pattern in humans where growth and motor skill development proceed from the head downward through the body.

Encephalitis

An inflammation of the brain, often caused by viral infection, that can lead to symptoms such as fever, confusion, seizures, and headaches.

Cephalopedal Trend

A principle of growth and development where development progresses from the head (cephalo) toward the feet (pedal).