Surely this is not what you meant to write! you meant, I think, that "let n= k+1, then n2= (k+1)2= k2+ 2k+ 1mamma_mia66 said:Homework Statement
Prove by induction that n2>(4n+7) for all integers n[tex]\geq[/tex]6.
[Hint: somewhere you may use the fact that (2k+1)[tex]\geq[/tex]13 > 4.]
Homework Equations
The Attempt at a Solution
n2> (4n+7)
(B) when n =6
62> (4*6+7)
36>31 true
(I) given n=k then k2> (4k+7)
let n=k+1= k2+2k+1
6k+8 > 4(k+1)+7
Is that looks like okay? Please help. Thank you.
To use induction, we first need to prove that the statement is true for a base case, which in this case is n=6. Then, we assume that the statement is true for some arbitrary value of n, and use that assumption to prove that it is also true for n+1. This allows us to conclude that the statement is true for all values of n≥6.
In order to use induction, we need to have a base case that is true. In this case, n=6 is the smallest value that makes the statement true. If we were to include smaller values of n, the statement would not hold and we would not be able to use induction to prove it.
Yes, for this statement, we can prove the base case n=6 by substituting n=6 into the statement and simplifying to get 36>31, which is true. Then, assuming the statement is true for some n, we can substitute n+1 into the statement and use algebraic manipulation to show that it is also true for n+1. This completes the proof by induction.
Yes, by using induction, we have proven that the statement holds true for all values of n≥6. This is because we have shown that if the statement is true for some value of n, it is also true for the next value n+1, and so on. Therefore, it will always be true for any value of n≥6.
Yes, there are other methods such as direct proof, proof by contradiction, or proof by contrapositive. However, induction is a commonly used method for proving statements that involve a pattern or sequence, like in this case where we need to show that the statement is true for all values of n≥6.