Proving Inequalities in Mathematical Induction

[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?

about the second question:
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?
`

 

[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.

 

[VietDao29]

1) Prove that:

[complete solution edited out]

NO COMPLETE SOLUTION!

 

[vanesch]

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

 

[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

 

[MathematicalPhysicist]

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