Using principle of mathematical induction, prove the following `1+3+5+

1.	Using principle of mathematical induction, prove the following `1+3+5+...+(2n - 1)=n^2`
Answer» Let the given statement be P (n). Then, ` P(n) : 1+3+5+7+…+(2n-1)= n^(2) ` Putting n = 1 in the given statement, we get LHS = 1 and RHS = ` 1^(2) = 1`. `:. ` LHS = RHS. Thus, P(1) is true. Let P (k) be true. Then, `P(k) : 1+3+5+7+...+(2k-1) = k^(2)`. ....(i) Now, ` 1+3+5+7+...+(2k-1)+{2(k+1)-1}` ` = {1+3+5+7+...+(2k-1)}+(2k+1)` ` =k^(2) (2k+1)" "` [using (i)] ` = (k+1)^(2)` ` :. P(k+1): 1+3+5+7+...(2k-1)+{2(k+1)-1}=(k+1)^(2)` This shows that P(k+1) is true, whenever P(k) is true. Thus, P(1) is true and P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction, we have `1+3+5+7+...+(2n-1)= n^(2) " for all " n in N`.

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