Section 1.5 « Difference Equations to Differential Equations

Section 1.5

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1. (a) $w_{n+1} = w_n + 0.00003w_n(1000 - w_n)$ , $n = 0, 1, 2, \ldots$

(c) The populatoin will double in $42$ years and triple in $102$ years.

(d) From the plot, it appears that $\displaystyle{\lim_{n \to \infty}w_n = 1000}$

3. (a) $p_{n+1} = 1.045p_n$ , $n = 0, 1, 2, \ldots$

(c) The population will double in $16$ years and triple in $25$ years.

(d) $\displaystyle{\lim_{n \to \infty}p_n = \infty}$

(e) $52$ years

4. (a) It appears that $\displaystyle{\lim_{n \to \infty}r_n = 10,000}$

(c) There appears to be a limiting cycle of period $2$ , with values alternating between $6,403$ and $11,930$ .

(e) There appears to be a limiting cycle of period $8$ , with values cycling thorough $4851$ , $11245$ , $7661$ , $12248$ , $5198$ , $11588$ , $6876$ , and $12375$ .

7. (a) $\displaystyle{\lim_{x \to \infty}x_n = \infty}$

(c) It appears that $\displaystyle{\lim_{n \to \infty}x_n = 0.7391}$ to four decimal places.

(e) It appears that $\displaystyle{\lim_{n \to \infty}x_n =\sqrt{2}}$ .

(g) It appears that $\displaystyle{\lim_{n \to \infty}x_n = \infty}$ .

(i) There appears to be a limiting cycle of order $2$ .

(k) The limiting behavior, if any, is not clear.

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