Question

13. Consider the recursive formula19751157u = 450un = 0.75un-1 + 210 where n 21a. Find uy, Un, Uz, UMandUgob. How can you calculate backward from the value of u, to Uy, or450? In general, what operations can you perform to any termin order to find the value of the previous term?c. Write a recursive formula that generates the values from 13a inreverse order.

Answer 1

See Explanation below

Step-by-step explanation:

a) The given Recursive formula:

`\begin{gathered} u_n=0.75u_(n-1)\text{ + 210} \\ \text{where u}_n\text{ = 450} \end{gathered}`

To get u1, n = 1

`\begin{gathered} u_1=0.75u_(1-1)+\text{ 210} \\ u_1=0.75u_0\text{ + 210} \\ u_1=\text{ 0.75}*450\text{ + 210} \\ u_1\text{ = 547.5} \end{gathered}`

To get u2, n = 2

`\begin{gathered} u_2=0.75u_(2-1)\text{ + 210} \\ u_2=0.75u_1\text{ + 210} \\ u_2\text{ = 0.75(547.5) + 210} \\ u_2\text{ = 410.625 + 210 } \\ u_2\text{ = 620.625} \end{gathered}`

We will follow the above steps to find u3, u4 and u5

b) To find the value of the previous term:

New term = un, previous term = un-1

`\begin{gathered} u_n=0.75u_(n-1)+\text{ 210 Make the previous term the subject of formula} \\ 0.75u_(n-1)\text{ = }u_n\text{ - 210 Divide through by 0.75} \\ u_(n-1)\text{ = }((u_n-210))/(0.75) \end{gathered}`

The order of operation is to make the previous term the subject of the formula.

This done by bringing 210 to the other side of the equation: subtract 210 on both sides of the equation.

Then dividing both sides of the equation by the coefficient of the previous equation (0.75).

c) A recursive formula that generates the values from 13a in

reverse order has been derived in question (b)

`u_(n-1)\text{ = }((u_n-210))/(0.75)`