Prove the following statement by mathematical induction. For every integer n2 0, 2" < (n + 2)!. Proof (by mathematical induction): Let P(n) be the inequality 2" < (n + 2)!. We will show that P(n) is true for every integer n 2 0. Show that P(0) is true: Before simplifying, the left-hand side of P(0) is 2 and the right-hand side is The fact that the statement is true can be deduced from that fact that 2° = 1. Show that for each integer k2 o, if P(k) is true, then P(k + 1) is true: Let k be any integer with k z 0, and suppose that P(k) is true. In other words, suppose that 2* < (*+ 2)! [This is P(k), the inductive hypothesis.] We must show that P(k + 1) is true. P(k + 1) is the inequality < (k+ 3)! Information about P(k + 1) can be deduced from the following steps. Identify the reason for each step. 2* < (k + 2)! 2*.2 < (k + 2)! - 2 2* +1 < (k + 2)! - 2 2* +1 < (k + 2)! - (k + 3) 2* +1 < (k + 3)! by the induction hypothesis v by basic algebra by basic algebra because 2
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The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
Please what is the answer of the left-hand side of P(0) is and the right-hand side is? And please show it clearly
![Prove the following statement by mathematical induction.
For every integer n2 0, 2" < (n + 2)!.
Proof (by mathematical induction): Let P(n) be the inequality 2" < (n + 2)!. We will show that P(n) is true for every integer n 2 0.
Show that P(0) is true: Before simplifying, the left-hand side of P(0) is 2
and the right-hand side is 6
The fact that the statement is true can be deduced from that fact that 2° = 1
Show that for each integer k2 0, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 0, and suppose that P(k) is true. In other words, suppose that 2* < (k+ 2)!
[This is P(k), the inductive hypothesis.]
We must show that P(k + 1) is true. P(k + 1) is the inequality
28+1 < (k+ 3)!
Information about P(k + 1) can be deduced from the following steps. Identify the reason for each step.
2k < (k + 2)!
2*.2 < (k + 2)! - 2
zk +1 < (k + 2)! - 2
2k +1 < (k + 2)! - (k + 3)
2k +1 < (k + 3)!
by the induction hypothesis v
by basic algebra
by basic algebra
because 2 <k + 3
by basic algebra
Since 2k +1« (k + 3)! the inequality for P(k + 1) is (true
Vv , which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc357075-d330-4116-93bc-b2819b26405c%2F10cd96f5-c1d6-4f7a-ab15-b15afb6a2e83%2F1gcvdo_processed.png&w=3840&q=75)
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