If A squared plus A plus 1=0, then A's 2009 power plus A's 2008 power plus... Add A and 2009

If A squared plus A plus 1=0, then A's 2009 power plus A's 2008 power plus... Add A and 2009

A's 2009 plus A's 2008 plus... Add A and 2009
=(A square plus A plus 1)(A 2007+ A 2004+--------- A 3+1)+2008
=2008

What is the last digit of the sum of the 100th power of 2 and the 101th power of minus 3?

Okay, we're only thinking in the last digits.
The first power of 2 is 2 The second power of 2 is 4 The third power of 2 is 8 The fourth power of 2 is 6
The fifth power of 2 is 2 The 6th power of 2 is 4 The 7th power of 2 is 8 The 8th power of 2 is 6
So every four-power cycle gives 2 to the 100th power of 6
In the same way
The first power of -3 is -3. The second power of -3 is 9. The third power of -3 is -7. The fourth power of -3 is 1.
The fifth power of -3 is -3. The sixth power of -3 is 9. The seventh power of -3 is -7. The eighth power of -3 is 1.
So every fourth power, one cycle leads to a 101th power of -3.