Click to Chat

1800-1023-196

+91-120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
`        For every integer n > 1, the inequality (n1)1/n `
5 years ago Navjyot Kalra
654 Points
```
Consider n numbers, namely 1, 2, 3, 4, . . . . . . n.

KEY CONCEPT: Now using A. M. > G. M. for distinct numbers, we get

1 + 2 + 3 + 4 . . . . . . . + n/n > (1. 2. 3. 4 . . . . n )1/n

⇒ n(n + 1)/2n > (n!)1/n ⇒ (n!)1/n < n + 1/2 ∴ True

ALTERNATE SOLUTION :

The given inequality can be written as

N! < (n + 1/2)n n > 1.

Let us use mathematical induction to check the validity of given inequality.

For n = 2, we have 2! < (3/2)2 = 9/4 which is true

∴ Inequality is valid for n = 2.

Let it be valid for n = k then k ! < (k + 1/2)k . . . . . . . . . . . . . . (1)

Consider

(k + 1)! = (k + 1) k! < (k + 1) ( k + 1/2)k (Using (1))

Now we will try to check

(k + 1) (k + 1/2)k < (k + 2/2)k + 1 . . . . . . . . . . . . . (2)

Which is equivalent to write 2 < (k + 2/k + 1)k + 1 . . . . . . . . . . . . (3)

Now, (k + 2/k + 1)k + 1 = (1 + 1/k + 1)k + 1

= 1 + (k + 1) 1/k + 1 + (k + 1)k/2! (1/k + 1)2 + . . . . . . . .(Using Binomial expansion)

∴ (k + 2/k + 1)k + 1 > 2 ⇒ (3) holds and hence (2) holds

⇒ (k + 1)! < (k + 2/2)k +1 . Thus statement is true for

n = k + 1

Hence, by Principle of Mathematical Induction given statement is true, ∀ n > 1.

```
5 years ago
Think You Can Provide A Better Answer ?

## Other Related Questions on Algebra

View all Questions »  ### Course Features

• 731 Video Lectures
• Revision Notes
• Previous Year Papers
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Test paper with Video Solution  ### Course Features

• 101 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions