Why is it that $2^{10} + 2^{9} + 2^{8} + \cdots + 2^{3} + 2^2 + 2^1 = 2^{11} - 2$? Why is it that $$2^{10} + 2^{9} + 2^{8} + \cdots + 2^{3} + 2^2 + 2^1 = 2^{11} - 2?$$

Try looking at the terms in binary:
  2^10   10000000000
  2^9     1000000000
  2^8      100000000
  2^7       10000000
  2^6        1000000
  2^5         100000
  2^4          10000
  2^3           1000
  2^2            100
+ 2^1             10
---------------------
=        11111111110

+   2             10
=       100000000000 = 2^11

