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Homework Statement
Show that the statement holds for all positive integers n.
1 + 3 + 5 + ... + (2n -1) = n2
Homework Equations
The Attempt at a Solution
Assume that k will work, then k + 1:
1 + 3 + 5 + ... + (2(k+1) -1) = (k+1)2
1 + 3 + 5 + ... + 2k+1 = k2 + 2k + 1
Recall that for k,
1 + 3 + 5 + ... + (2k -1) = k2
Then k+1,
1 + 3 + 5 + ... + (2k -1) + (2k+1) = k2 + (2k + 1)
Is this enough to conclude that the statement holds?