Prove Inequalities using Mathematical Induction examples with solutions
Question 1:
Prove the following inequality by the principle of mathematical induction:
, n ∈ N.
Solution:
We can write,
Step 1:
We first show that the basis for induction P(1) is true, that is, P(n) is true for .
Left Side ,
Right Side .
Since, , that is, Left Side > Right Side
Hence, P(1) is true, that is, P(n) is true for .
Step 2:
Assume that P(k) is true for some natural number k, that is,
We need to prove that P(k +1) is also true.
Because P(k) is true, so we have,
Add to both sides:
.
That is, we get
.
Thus, P(k +1) is true whenever P(k) is true.
Therefore, from the principle of mathematical induction, the statement:
is true for all natural number n.
Question 2:
Prove the following statement for all integers by the principle of mathematical induction:
.
Solution:
We can write,
Step 1:
We first show that the basis for induction P(4) is true, that is, P(n) is true for .
Left Side ,
Right Side .
Since, , that is, , that is, Left Side > Right Side
Hence, P(4) is true, that is, P(n) is true for .
Step 2:
Assume that P(k) is true for all integers , that is,
We need to prove that P(k +1) is also true, that is, we need to prove that if .
To prove required from this, consider,
(Apply by the induction hypothesis)
(Because so )
.
That is, we get
.
Thus, P(k +1) is true whenever P(k) is true for all integers .
Therefore, from the principle of mathematical induction, the statement:
is true for all integers .
Tags: mathematical induction examples with solutions, prove inequalities by induction
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