Question

Prove by mathematical induction that 1+3+5+7+......+(2n-1)=n²

Answer 1

Proof by induction takes two steps.
Step 1: Show that the left hand side equals the right hand side for n = 1
LS... 2(1) - 1 = 1
RS... 1^2 = 1
So step 1 holds true, thus meaning we can proceed to step 2 the induction step:
Assume true for n=k and show true for n=k+1
Show that the LS = RS
1+ 3+5... + (2k-1) + {2(k+1) - 1} = (k+1)^2
Now we know that 1+3+... (2k-1) is equal to k^2 so replace this one the left side
k^2 + {2(k+1) - 1} = (k+1)(k+1)
Now simplify both sides by expanding...
You will see that the LS = RS thus 1+ 3 + 5 +... = n^2