Question

Consider the table:
1 = 0 + 1
2 + 3 + 4 = 1 + 8
5+ 6+ 7+8 +9 + 8 + 27
10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64
Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it.

Answer 1

0

Answer 2

First , in the third line of your post, you need 9 instead of 27. the pattern is ((n)^2 +1 ) + ((n^2)+2) ...(n+1)^2 = n^3 + (n+1)^3
Use algebra to demonstrate this equivalence.

Answer 3

The third line should be 5 + 6 + 7 + 8 + 9 = 8 + 27
Each row add up to the sum of the last cube and the next consecutive cube. But I don't see how to write the expression.