Prove the statement is true using mathematical induction: 2n-1 ≤ n! Use the space below to write your answer. To make the < symbol, you might want to use the < with the underline feature.

Question

Ryan

Mathematics

13 September, 03:05

Prove the statement is true using mathematical induction: 2n-1 ≤ n! Use the space below to write your answer. To make the < symbol, you might want to use the < with the underline feature.

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Answers (1)

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Johanna Gutierrez · Answer 1

Step-by-step explanation:

Given that n! = n (n - 1) (n - 2) (n - 3) ... 2*1

We want to show that 2n - 1 ≤ n!

Since

n! = n (n - 1) (n - 2) (n - 3) ... 2*1

= n (n - 1) !

n! = n (n - 1) (n - 2) !

n! = (n² - n) (n - 2) !

From here obviously,

n! ≥ n

n! ≥ 2n

And

n! ≥ 2n - 1

Which implies

2n - 1 ≤ n!