SOLUTION: Find the 25th term in the sequence 0, 3, 8, 15, 24, 35, ..........

Algebra ->  Sequences-and-series -> SOLUTION: Find the 25th term in the sequence 0, 3, 8, 15, 24, 35, ..........       Log On


   

Question 1049661: Find the 25th term in the sequence 0, 3, 8, 15, 24, 35, ..........
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

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+%2B+3+=+4
1+%2B+3+%2B+5+=+9
1+%2B+3+%2B+5+%2B+7+=+16.....and so on.
The formula for this is
1+%2B+3+%2B+5 + ... ++%282n-1%29+=+n%5E2
Your sequence is just my sum with 1 removed from the start! So the
formula is:
a%5Bn%5D+=+n%5E2+-+1 (for all terms given)

then the 25th term in the sequence is:

a%5B25%5D+=+25%5E2+-+1
a%5B25%5D+=+625+-+1
a%5B25%5D+=+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}....