# Two inequality questions.

1. Feb 13, 2006

### MathematicalPhysicist

1) provethat:
n n
sum(a_k)+1<= product(1+a_k)
k=1 k=1
when a_k>0 for every k natural, or when -1<a_k<0

2) x1,...x_n>0
n>1 x1x2..x_n=1
prove by induction on n that x1+x2+...+x_n>n

concerning the first question i tried to open the product this way:
n
product(1+a_k)=(1+a1)(1+a2)...(1+an)=1+a1(a2+..an)+a1a2...an+a2(a3+...+an)+a3(a4+...an)+an
from here its apparent that it's greater than the sum, is my opening correct?

i have these two:
x1+x2+...xn>k
k-1+x1+x2+..+xn>2k-1>=k+1
then i only need to prove that:
x1+..+xk+1>k-1+x1+...+xk
or:
xk+1>k-1
if we use this: x1x2...xkxk+1=xk+1
we get:
x1x2...xkxk+1>k-1
now how do i approach it from there on?
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2. Feb 13, 2006

### MathematicalPhysicist

well induction is fine by me, but have i opned the product correctly, because if i have it's self apparent that it's bigger or equals the sum.

btw, what about the second question?

thank you for your help, induction does look much simpler than my approach.

3. Feb 13, 2006

### VietDao29

NO COMPLETE SOLUTION!!!!!!!!!!!!!!!!!!!:grumpy: :grumpy: :grumpy: :grumpy: :grumpy:

Last edited by a moderator: Feb 13, 2006
4. Feb 13, 2006

### vanesch

Staff Emeritus
benorin, I've deleted your post because it contained a complete solution to the question asked (even without any pedagogical explanation).

5. Feb 13, 2006

### benorin

FYI

$$\prod_{k=1}^{n} (1+ a_k z) = 1 + \sum_{q=1}^{n} z^{q} \left[ \sum_{1 \leq p_1 < p_2 < \cdots < p_q \leq n} \left( \prod_{k=1}^{q} a_{p_k} \right) \right]$$

put z=1 and verify.

Thanks, I need to work that one out myself

--Ben

6. Feb 14, 2006

### MathematicalPhysicist

what about my second question? can i get some hints?