jmc

Question

Let S = 2^0,2^1,2^2,,2^10. Consider all possible positive differences of pairs of elements of S. Let N be the sum of all of these differences. Find N.

Accepted Answer

When computing N, the number 2^x will be added x times (for terms 2^x-2^0, 2^x-2^1, , 2^x - 2^x-1), and subtracted 10-x times. Hence, N can be computed as N=10 2^10 + 8 2^9 + 6 2^8 + - 8 2^1 - 10 2^0.Then align* N & = 10(2^10-1) + 8(2^9 - 2^1) + 6(2^8-2^2) + 4(2^7-2^3) + 2(2^6-2^4) & = 10(1023) + 8(510) + 6(252) + 4(120) + 2(48) & = 10(1000+23) + 8(500+10) + 6(250+2) + 480 + 96 &= 16398. align*