Prove That $$\sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { for all positive integers } n$$

question 36

Essay

Prove that $$\sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { for all positive integers } n$$

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Definitions:

Idiosyncratic Reaction

An unusual or unexpected reaction to a drug that is not related to the dose or duration of exposure.

Toxic Effect

Harmful physiological effects caused by exposure to toxic substances or chemicals.

Controlled Substances

Drugs or chemicals whose manufacture, possession, and use are regulated by the government due to their potential for abuse or addiction.

Unused Quantities

Items or substances not yet utilized or consumed.