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An example Suppose you get a 5-year certificate of deposit (CD) at your local bank for $100000, and it earns interest at the annualized rate of exactly 5 percent per year The CD will be worth $127628 after 6 years To calculate this, multiply $1000 by 105, then multiply this result by 105, and repeat this process a total of 5 times The resulting numbers form a geometric sequence:

Once we deliver the product or service, it just begins to show its value. It may be months sometimes even years before the value of the new product or service can be fully realized. Yet true success of the project is measured by that value being realized during those years in production. We should arrange for a post-project business review evaluating whether the value is being realized and the project is showing the expected return on investment. In addition to this review, we can also do two other things. One is to ensure the satisfaction of peripheral stakeholders. The other is to ensure that the customer knows that value is being realized, and recognizes that value and gives credit to the project and the project team.

After 1 year: $100000 105 = $105000 After 2 years: $105000 105 = $110250 After 3 years: $110250 105 = $115763 After 4 years: $115763 105 = $121551 After 5 years: $121551 105 = $127628

Another example Is the following sequence a geometric sequence If so, what are the values t0 (the starting value) and k (the factor of change)

T = 3, 6, 12, 24, 48, 96, This is a geometric sequence The numbers change by a factor of 2 In this case, t 0 = 3 and k = 2

Achieving legal and nancial closure is often but not always a simple matter. Were all payments delivered Were they received and recorded Is there a letter on le recognizing successful completion of each contract, not only for the project as a whole, but also with each vendor For small projects, we may need to ask whether all warranty cards were sent to the manufacturers. Did everything get delivered, and was it received The best approach to ensuring that the paperwork le is complete is to make a checklist, and then make friendly contact with stakeholders to ensure that the paperwork is done. Sometimes, however, di culties, disagreements, procrastination, lost items, and forgetfulness get in the way. These problems do come up, and that is exactly why this work is important. If these issues are present, we nd them. And, by taking a direct and cooperative approach, we can almost always resolve them before they expand out of control, using up time and money. In my experience, this is much easier to do while people still remember the project, rather than later, when they have put it out of their minds. It is possible to come to good legal and nancial closure even when a project fails or delivers only partial results. If we are able to maintain a no-blame environment throughout the project, even when parts of it are falling apart, then usually everyone will see it is in each person s best interest to bring things to closure in a fair way.

5.4 A lever is driven by a bidirectional constant-speed motor to a position determined by a three-state controller. The motor has a speed of 10 percent of full stroke per second, and an inertial time constant of 1.0 sec. Differential gap in the controller is 2 percent of full stroke. How wide does the dead zone have to be to prevent limit cycling What would be the period of the cycling 5.5 A batch chemical reactor is to be brought up to operating temperature with a dual-mode system. Full controller output supplies heat through a hotwater valve, while zero output opens a cold-mater valve fully; at 50 percent output, both valves are closed. While full heating is applied, the temperature of the batch rises at, l F/min; the time constant of the jacket is estimated at 3 min, and the total dead time of the system is 2 min. The normal load is equivalent to 30 percent, of controller output. Estimate the required values for the three adjustments in the optimal switching program. 5.6 h given linear process is undamped with a proportional band setting of 50 percent for a linear controller. If a continuous nonlinear controller is used with a linearity setting of /3 = 0.2, how narrow can the proportional band be set and still tolerate an error of 20 percent