Prove the following: for all n ≥ 1,
Xn
i=1
i
!2
=
Xn
i=1
i
3
.
Answers
Answer:
ANSWER
If x
1
x
2
= 1, then
x
1
+x
2
=(
x
1
−
x
2
)
2
+2
x
1
x
2
≥2
x
1
x
2
=2,
so that the inequality holds true for n = 2.
Now assume that the sum of any m positive numbers whose product is 1 is greater than or equal to m and let x
1
,x
2
,....,x
m
,x
m+1
be (m + 1) positive integers such that
x
1
.x
2
...x
m
.x
m+1
=1
We shall prove
x
1
+x
2
+...++x
m
+x
m+1
=1
We shall prove
x
1
+x
2
+...+x
m
+x
m+1
≥m+1.
If each x
i
(i = 1,2,...,m + 1) is 1, then x
1
+x
2
+...x
m
+x
m+1
=m+, so that in this case the inequality holds true.
If x
i
′
s are not all 1, then among them there will be a number greater than 1 and a number less than 1. Let
x
m
> 1 and x
m+1
< 1.
We have x
1
x
2
...x
m−1
(x
m
x
m+1
) = 1.
This is a product of m numbers and so by our induction hypothesis, we can say that
x
1
+x
2
+...x
m−1
x
m
x
m+1
≤m
But then
x
1
+x
2
+...x
m−1
+x
m
+x
m+1
≤m−x
m
x
m+1
+x
m
+x
m+1
by
=m+1+(x
m
−1)(1−x
m+1
)>m+1
Since x
m
-1>0 and 1 - x
m+1
>0 by
so that (x
m
−1)(1−x
m+1
)>0.
This completes the proof.