minimumunstick;
N1 will be shown as a percentage of the fan maximum RPM.
Not true. RPM's are displayed as a percentage of some arbitrary reference RPM which, in general, is not the maximum RPM. The maximum RPM varies with the rating, e.g.: TOGA, Max. Continuous, Max Climb, Max Cruise.
If I set a certain N1 setting for take off and do not move the thrust levers as I climb, what will the N1 display show (assuming no autothrottle etc.)?
That depends on the display. In most cases it will show actual N1 increasing as the engine spools up, settling at the N1 that
corresponds with the thrust lever position (assuming that your engine is N1-controlled).
If I am not mistaken it will decrease as we go up in altitude, meaning that the fan RPM will goes down), which means that we have to move the thrust levers forward to keep our maximum fan speed or N1 setting.
That depends on what you have selected and how the engine is controlled
(i.e. engine control system design). If you selected CLB and that rating is implemented in the EEC, then the RPM's will be whatever is necessary to provide the rated thrust schedule as you climb. If you selected a throttle setting below max, then some engines will maintain N1 and other engines will maintain N2 as you climb, until you hit some engine limit, and then (typically) the control stays on that limit. If you are climbing with constant N2, then N1 may increase or decrease, depending on the engine's thermodynamic characteristics. For a given engine, the relation between N1 and N2 depends on ambient temperature, not density.
EDIT: It may also be affected by change of Mach number as you climb.
the engine will not be able to produce the same amount of thrust due to the thinner air.
See previous posts.
regards,
HN39
P.S. Density depends on pressure and temperature. It is usual to treat these parameters separately, since they have distinctly different effects on engine performance, as explained in earlier posts.