A Sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$

question 19

Essay

A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined as follows:
$$a _ { 1 } = 3 , \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 2 \quad \text { for all integers } k \geq 2 \text {. }$$
(a) Find $a _ { 1 } , a _ { 2 }$ , and $a _ { 3 }$ .
(b) Supposing that $a _ { 5 } = 4 ^ { 4 } \cdot 3 + 4 ^ { 3 } \cdot 2 + 4 ^ { 2 } \cdot 2 + 4 \cdot 2 + 2$ , find a similar numerical expression for $a _ { 6 }$ by substituting the right-hand side of this equation in place of $a _ { 5 }$ in the equation
$$a _ { 6 } = 4 \cdot a _ { 5 } + 2 .$$
(c) Guess an explicit formula for $a _ { n }$ . Simplify your answer using one of the following reference formulas:
$1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ .
$1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ .

---

Definitions:

Net Capital Outflow

The difference between a country's purchase of foreign assets and the sale of domestic assets to foreigners in a given time period.

Real Exchange Rate

The nominal exchange rate adjusted for differences in price levels between two countries, showing how many units of a foreign product can be purchased with one unit of a domestic product.

Foreign Direct Investment

An investment made by a company or individual in one country in business interests in another country, in the form of either establishing business operations or acquiring business assets.

Net Capital Outflows

The difference between the purchase of foreign assets by domestic residents and the purchase of domestic assets by foreigners, indicating the flow of capital from a country.