Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`1 + 2 + 3 + ...+ n
Answer» Let the given statement be P(n) . Then, ` P(n): (1+2+3+…+n) lt 1/8 (2n+1)^(2)`. When n =1, we have LHS = 1 and RHS = `1/8 (2 xx 1+1)^(2) = 9/8`. Clearly, ` 1 lt 9/8`. `:. ` P(1) is true. Let P(k) be true. Then, `P(k): (1+2+3+...+k) lt 1/8 (2k+1)^(2)`. ....(i) Now, `1+2+3+...+k+(k+1)` ` = {1+2+3+...+k} + (k+1)` ` lt 1/8 (2k+1)^(2) +(k+1) = ((2k+1)^(2)+8(k+1))/8 ` [using (i)] ` = ((4k^(2)+12k+9))/8 = ((2k+3)^(2))/8 = ({2(k+1)+1}^(2))/8 ` . `:. {1+2+3+...+k(k+1)} lt ({2(k+1)+1}^(2))/8 `. This shows that P(k+1) is true. Thus, P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction, it follows that `(1+2+3+...+m) lt 1/8 (2n+1)^(2) " for all values of " n in N`.

Discussion