• Support PF! Buy your school textbooks, materials and every day products Here!

Having trouble proving stuff

  • Thread starter synkk
  • Start date
  • #1
216
0
Prove for all ## x \geq -1 ## and n being a natural number, ## (1+x)^n \geq 1 + nx ## I've done this using induction

hence deduce from this that for any real number # a \ geq 2 ##, ## a^n > n ## for all n belonging to the natural numbers.

I'm stuck at the deduce part

I'm not sure what deduce means, it seems fairly obvious so instead I tried to prove it, by again, induction:

a = 2,
2^n > n which is true a n >= 1
assume true for a = k

then when a = k+1

(k+1)^n > n

I don't see how I can prove it from here, it seems obvious as k >= 2 and n >=1 this will always hold true, and thus a^n > n but I can't seem to prove it. Does anyone have any tips on proving as I'm fairly new at this

thank you
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,740
412
Prove for all ## x \geq -1 ## and n being a natural number, ## (1+x)^n \geq 1 + nx ## I've done this using induction

hence deduce from this that for any real number # a \ geq 2 ##, ## a^n > n ## for all n belonging to the natural numbers.

I'm stuck at the deduce part

I'm not sure what deduce means
It means "show that this follows from what you have just proved".

In this case, you've shown that if [itex]x \geq -1[/itex] then [itex](1 + x)^n \geq 1 + nx[/itex].

Now see what happens if you set [itex]a = 1 + x[/itex].
 
  • #3
216
0
It means "show that this follows from what you have just proved".

In this case, you've shown that if [itex]x \geq -1[/itex] then [itex](1 + x)^n \geq 1 + nx[/itex].

Now see what happens if you set [itex]a = 1 + x[/itex].
how would that work exactly? it states that a >= 2 if we let a = 1 + x then a >= 0 as x >= -1?
 
Last edited:
  • #4
216
0
if I ignore my confusion above I ge tthis:

a^n >= 1 + nx

as x >= -1 a^n >= 1 - n

as n is a natural number n >=1
hence a^n > n
 

Related Threads on Having trouble proving stuff

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
3
Views
1K
Replies
5
Views
1K
Replies
8
Views
1K
Replies
22
Views
9K
Replies
4
Views
2K
Replies
2
Views
1K
Replies
4
Views
1K
Replies
5
Views
3K
Top