What is the sum of the first n odd numbers? (The first five odd numbers are 1, 3, 5, 7, 9.)
1 = 1 = 12
1 + 3 = 4 = 22 = ((1 + 3)/2)2
1 + 3 + 5 = 9 = 32 = ((1 + 5)/2)2
1 + 3 + 5 + 7 = 16 = 42 = ((1 + 7)/2)2
So 1 + 3 + 5 + … + n = ((1 + n)/2)2
1 + 3 = 4 = 22 = ((1 + 3)/2)2
1 + 3 + 5 = 9 = 32 = ((1 + 5)/2)2
1 + 3 + 5 + 7 = 16 = 42 = ((1 + 7)/2)2
So 1 + 3 + 5 + … + n = ((1 + n)/2)2
OR
1 = 1 = (0 + 1)2
1 + 3 = 1 + (2 · 1 + 1) = 4 = (1 + 1)2
1 + 3 + 5 = 1 + 3 + (2 · 2 + 1) = 9 = (2 + 1)2
1 + … + 7 = 1 + … + (2 · 3 + 1) = 16 = (3 + 1)2
So 1 + 3 + … + (2k + 1) = (k + 1)2
1 + 3 = 1 + (2 · 1 + 1) = 4 = (1 + 1)2
1 + 3 + 5 = 1 + 3 + (2 · 2 + 1) = 9 = (2 + 1)2
1 + … + 7 = 1 + … + (2 · 3 + 1) = 16 = (3 + 1)2
So 1 + 3 + … + (2k + 1) = (k + 1)2