Question

Determine the sum of the first k odd positive integers for a number of values of k. What generalizations occur to you? Are your inferences correct for all positive integers k?

Answer 1

Let be
`n_i` the i-th odd positive integer number,
`\sum_(i=1)^(k){n_i}=k^2`

Explanation:

Let be
`n_i` the i-th odd positive integer number

`k=1\\n_1=1\\S_1=n_1=1\\\\k=2\\n_1=1, n_2=3\\S_2=n_1+n_2=1+3=4\\\\ k=3\\n_1=1, n_2=3, n_3=5\\S_3=n_1+n_2+n_3=1+3+5=9\\\\\\k=4\\n_1=1, n_2=3, n_3=5, n_4=7\\S_4=n_1+n_2+n_3+n_4=1+3+5+7=16\\\\`

It seems that the sum of the first k odd positives integers is
`k ^ 2`

Note that,

`n_1=1=1+2* 0\\n_2=3=1+2* 1\\n_3=5=1+2* 2\\n_4=7=1+2* 3\\n_5=9=1+2* 4\\.\\.\\.\\n_k=2k-1=1+2* (k-1)\\`

thus,

`\sum_(i=1)^(k){n_i}=\sum_(i=1)^(k){(2i-1)}=2\sum_(i=1)^(k)i-\sum_(i=1)^(k)1\\\\\sum_(i=1)^(k){n_i}=2(k(k+1))/(2)-k=k^2`