Inequality (1 Viewer)

lpodtouch · Aug 13, 2015

Could someone please help me with the last part of

this inequality qn (worth 2 marks):

InteGrand · Aug 13, 2015

lpodtouch said:
Could someone please help me with the last part of View attachment 32285this inequality qn (worth 2 marks):

$If we expand the L.H.S., we will obtain $n$ terms which are each 1, corresponding to terms when we multiply $a_1$ by $\frac{1}{a_1}$, $a_2$ by $\frac{1}{a_2}$, etc. The remaining terms will be of the form $\frac{a_i}{a_j}$, where $i\neq j$, and $i,j\in \{1,2,3,...,n \}$. For each term $\frac{a_i}{a_j}$, there is a corresponding reciprocal term $\frac{a_j}{a_i}$ (e.g., we obtain $\frac{a_1}{a_n}$ by multiplying the first term of the first bracket with the last term of the second bracket, and we obtain the reciprocal term $\frac{a_n}{a_1}$ by multiplying the last term of the first bracket with the first term of the second bracket). In total, there are $\binom{n}{2}=\frac{n(n-1)}{2}$ such reciprocal pairs, since there are $\binom{n}{2}$ ways to choose 2 non-identical integers $i,j$ from the set $\{1,2,3,...,n \}$. Since each reciprocal pair has sum greater than or equal to 2, we have$

$LHS=n\times 1 +\left(\text{sum of }\frac{n(n-1)}{2}\text{ reciprocal pair terms} \right)$

$\geq n +\frac{n(n-1)}{2}\times 2$

$$=n^2$.$

RealiseNothing · Aug 14, 2015

lpodtouch said:
Could someone please help me with the last part of View attachment 32285this inequality qn (worth 2 marks):

Alternatively, consider:

$\sum_{k=1}^{n} (a_kx+b_k)^2$

$(a_1^2+...+a_n^2)x^2+2(a_1b_1+...+a_nb_n)x+(b_1^2+...+b_n^2)$

This has discriminant $\leq0$

$4(a_1b_1+...+a_nb_n)^2-4(a_1^2+...+a_n^2)(b_1^2+...+b_n^2) \leq 0$

$(a_1b_1+...+a_nb_n)^2 \leq (a_1^2+...+a_n^2)(b_1^2+...+b_n^2)$

Let $a_k=\sqrt{x_k}$ and $b_k=\frac{1}{\sqrt{x_k}}$ so that we obtain:

$(1+1+...+1)^2 \leq (x_1+...+x_n)(\frac{1}{x_1}+...+\frac{1}{x_n})$

$(x_1+...+x_n)(\frac{1}{x_1}+...+\frac{1}{x_n}) \geq n^2$

simpleetal · Aug 14, 2015

Inequality (1 Viewer)

lpodtouch

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InteGrand

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RealiseNothing

what is that?It is Cowpea

simpleetal

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