Question 1150514
<br>
The number of cans in a pile with n cans on the bottom row is<br>
S = 1 + 2 + 3 + ... + (n-1) + n<br>
The sum of the integers from 1 to n is<br>
{{{S = (n(n+1))/2}}}<br>
(a) S = 3240<br>
{{{3240 = (n(n+1))/2}}}
{{{6480 = n(n+1)}}}<br>
Solve by inspection: 80(81) = 6480, so n = 80.<br>
(b) {{{n^2+n-2S = n^2+n-2((n(n+1))/2) = n^2+n-(n^2+n) = 0}}}<br>
(c) S = 2100<br>
{{{2100 = (n(n+1))/2}}}
{{{4200 = n(n+1)}}}<br>
The solution to that equation is not an integer, so there can't be that many cans in the pile.<br>