Question

Investigate the sums of consecutive odd numbers starting at 1. what do you notice?

Answer 1

The sum of n odd numbers starting at 1 is n²

Explanation:

The sum of n odd numbers starting at 1 is n²

Proof:

1 + 3 + 5 + 7 + ...

= 1 + (1 + 2) + (1 + 4) + (1 + 6) + ...

= 1 + 1 + .... n times + 2 + 4 + 6 + ... (n-1) times

`=\sum\limits^n_1 {1} + 2 ( 1 + 2 + 3 + ...(n-1) times)\\\\=n + 2\sum\limits^{^(i=n-1)}_{_(i=1)} {i} \\\\= n + 2((n-1)(n-1 + 1))/(2) \;\; (sum\;of\;n\;consecutive \;numbers\;is\;(n(n+1))/(2) )\\\\= n + (n-1)(n)\\\\= n(1+n-1)\\\\= n(n)\\\\=n^2`