Use the Principle of Mathematical Induction to Prove That $$1 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 0$$

question 4

Short Answer

Use the Principle of Mathematical Induction to prove that $$1 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 0$$

---

Definitions:

Marginal Social Cost

The total cost to society of producing an additional unit of a good, including both private costs and any externalities.

External Cost

Costs of a transaction that affect people other than the buyer or seller, often not included in the price of goods or services.

Efficient Amount

The level of production or quantity of a good that optimizes the use of resources to achieve the highest possible satisfaction or benefits.

Unregulated Amount

A quantity or volume not controlled or limited by laws or administrative rules, often relating to production or trade.