MHB Find the smallest A satisfying the inequality

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The discussion revolves around finding the smallest real number A that satisfies the inequality involving the Fibonacci sequence defined by a_1 = 1, a_2 = 1, and a_n = a_{n-1} + a_{n-2} for n > 2. The specific inequality to be satisfied is \[\sum_{i = 1}^{k}\frac{1}{a_{i}a_{i+2}} \leq A\] for any natural number k. Participants are tasked with determining the value of A that holds true for all k. The conversation highlights the mathematical approach to solving this inequality using properties of the Fibonacci sequence. Ultimately, the goal is to identify the smallest A that meets the criteria outlined in the inequality.
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Let $a_1 = 1$, $a_2 = 1$ and $a_n = a_{n-1}+a_{n-2}$ for each $n > 2$. Find the smallest real number, $A$, satisfying

\[\sum_{i = 1}^{k}\frac{1}{a_{i}a_{i+2}} \leq A\]

for any natural number $k$.
 
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lfdahl said:
Let $a_1 = 1$, $a_2 = 1$ and $a_n = a_{n-1}+a_{n-2}$ for each $n > 2$. Find the smallest real number, $A$, satisfying

\[\sum_{i = 1}^{k}\frac{1}{a_{i}a_{i+2}} \leq A\]

for any natural number $k$.

we have
$\frac{1}{a_{i}a_{i+2}}= \frac{1}{a_{i+1}}\frac{a_{i+1}}{a_{i}a_{i+2}}$
$= \frac{1}{a_{i+1}}\frac{a_{i+2}- a_i}{a_{i}a_{i+2}}$ (from given condition)
$= \frac{1}{a_{i+1}}(\frac{1}{a_i} - \frac{1} {a_{i+2}})$
above term is positive and telescopic sum so maximum sum is sum at infinite and adding we get
$\frac{1}{a_1a_2} + \frac{1}{a_2a_3}$ and se get putting the values $\frac{3}{2}$
 
kaliprasad said:
we have
$\frac{1}{a_{i}a_{i+2}}= \frac{1}{a_{i+1}}\frac{a_{i+1}}{a_{i}a_{i+2}}$
$= \frac{1}{a_{i+1}}\frac{a_{i+2}- a_i}{a_{i}a_{i+2}}$ (from given condition)
$= \frac{1}{a_{i+1}}(\frac{1}{a_i} - \frac{1} {a_{i+2}})$
above term is positive and telescopic sum so maximum sum is sum at infinite and adding we get
$\frac{1}{a_1a_2} + \frac{1}{a_2a_3}$ and se get putting the values $\frac{3}{2}$

Hi, kaliprasad

With your telescoping sum, I get:
\[\sum_{i=1}^{k}\frac{1}{a_{i}a_{i+2}} = \sum_{i=1}^{k}\left ( \frac{1}{a_{i}a_{i+1}}-\frac{1}{a_{i+1}a_{i+2}} \right )\\ =\frac{1}{a_{1}a_{2}}-\frac{1}{a_2a_3}+\frac{1}{a_2a_3}-...+ \frac{1}{a_{k-1}a_k}-\frac{1}{a_ka_{k+1}}+\frac{1}{a_ka_{k+1}}-\frac{1}{a_{k+1}a_{k+2}} \\ = \frac{1}{a_{1}a_{2}}-\frac{1}{a_{k+1}a_{k+2}} = 1-\frac{1}{a_{k+1}a_{k+2}}.\]

So the smallest possible $A$, that satisfies the inequality is:
\[\lim_{k\rightarrow \infty }\left ( 1-\frac{1}{a_{k+1}a_{k+2}} \right ) = 1.\]
 
lfdahl said:
Hi, kaliprasad

With your telescoping sum, I get:
\[\sum_{i=1}^{k}\frac{1}{a_{i}a_{i+2}} = \sum_{i=1}^{k}\left ( \frac{1}{a_{i}a_{i+1}}-\frac{1}{a_{i+1}a_{i+2}} \right )\\ =\frac{1}{a_{1}a_{2}}-\frac{1}{a_2a_3}+\frac{1}{a_2a_3}-...+ \frac{1}{a_{k-1}a_k}-\frac{1}{a_ka_{k+1}}+\frac{1}{a_ka_{k+1}}-\frac{1}{a_{k+1}a_{k+2}} \\ = \frac{1}{a_{1}a_{2}}-\frac{1}{a_{k+1}a_{k+2}} = 1-\frac{1}{a_{k+1}a_{k+2}}.\]

So the smallest possible $A$, that satisfies the inequality is:
\[\lim_{k\rightarrow \infty }\left ( 1-\frac{1}{a_{k+1}a_{k+2}} \right ) = 1.\]

it is my mistake.
Your telescopic sum and hence limit is right.
 
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