Question 1049661


Find the 25th term in the sequence {{{0}}}, {{{3}}}, {{{8}}}, {{{15}}}, {{{24}}},{{{ 35}}}, ..........

The sequence of differences is:

{{{0}}}, .....{{{3}}},..... {{{8}}},..... {{{15}}},..... {{{24}}},.....{{{ 35}}}, 
.....{{{3}}}......{{{5}}}........{{{7}}}.......{{{9}}}...........{{{11}}}->first  differences consecutive odd numbers
.........{{{2}}}.......{{{2}}}.......{{{2}}}...........{{{2}}}...->second differences same, so no need to go further

I happen to know that the sum of consecutive odd numbers is a square:

  {{{1 + 3 = 4}}}
  {{{1 + 3 + 5 = 9}}}
  {{{1 + 3 + 5 + 7 = 16}}}.....and so on.  

The formula for this is

  {{{1 + 3 + 5}}} + ... +{{{ (2n-1) = n^2}}}

Your sequence is just my sum with {{{1}}} removed from the start! So the 
formula is:

  {{{a[n] = n^2 - 1}}}  (for all terms given)


then the 25th term in the sequence is:


{{{a[25] = 25^2 - 1}}}

{{{a[25] = 625 - 1}}}

{{{a[25] = 624}}}


here it is:
{0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, {{{624}}}, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443}....