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 The sequence \((a_n)\) is defined by \(a_1=\frac{1}{2}\) and 

\(a_n = a_{n - 1}^2 + a_{n - 1}\)

for \(n \ge 2.\)

 

Prove that 

\(\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2\ \)

for all \(n \ge 1.\)

 

-------Thanks! laughcoolsmiley

 Apr 29, 2020
 #2
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+2

๐…๐ข๐ซ๐ฌ๐ญ, ๐ฒ๐จ๐ฎ ๐ฌ๐ก๐จ๐ฎ๐ฅ๐ ๐ ๐ž๐ง๐ž๐ซ๐š๐ญ๐ž ๐š๐›๐จ๐ฎ๐ญ ๐Ÿ๐ŸŽ ๐ญ๐ž๐ซ๐ฆ๐ฌ ๐จ๐Ÿ ๐ญ๐ก๐ž ๐Ÿ๐ข๐ซ๐ฌ๐ญ ๐ฌ๐ž๐ช๐ฎ๐ž๐ง๐œ๐ž ๐š๐ฌ ๐Ÿ๐จ๐ฅ๐ฅ๐จ๐ฐ๐ฌ:

 

๐ฅ๐ข๐ฌ๐ญ๐Ÿ๐จ๐ซ๐ž๐š๐œ๐ก(๐ง, ๐=(๐ŸŽ.๐Ÿ“, ๐ŸŽ.๐Ÿ•๐Ÿ“ , ๐Ÿ.๐Ÿ‘๐Ÿ๐Ÿ๐Ÿ“ , ๐Ÿ‘.๐ŸŽ๐Ÿ‘๐Ÿ“๐Ÿ๐Ÿ“๐Ÿ”๐Ÿ๐Ÿ“ , ๐Ÿ๐Ÿ.๐Ÿ๐Ÿ’๐Ÿ•๐Ÿ‘๐Ÿ๐Ÿ—๐Ÿ•๐Ÿ , ๐Ÿ๐Ÿ”๐Ÿ.๐Ÿ๐Ÿ’๐Ÿ’๐Ÿ’๐Ÿ๐Ÿ’๐Ÿ– , ๐Ÿ๐Ÿ”๐Ÿ’๐Ÿ–๐Ÿ“.๐Ÿ’๐Ÿ—๐Ÿ’๐Ÿ“๐Ÿ’ , ๐Ÿ•๐ŸŽ๐Ÿ๐Ÿ“๐ŸŽ๐Ÿ•๐Ÿ—๐ŸŽ๐Ÿ”.๐Ÿ• , ๐Ÿ’๐Ÿ—๐Ÿ๐Ÿ๐Ÿ๐Ÿ‘๐Ÿ‘๐Ÿ’๐Ÿ‘๐Ÿ–๐Ÿ•๐Ÿ‘๐Ÿ‘๐Ÿ‘๐Ÿ”๐Ÿ”๐Ÿ๐ŸŽ , ๐Ÿ๐Ÿ’๐Ÿ๐Ÿ๐Ÿ•๐Ÿ“๐Ÿ“๐Ÿ’๐Ÿ‘๐Ÿ๐Ÿ๐Ÿ–๐Ÿ๐Ÿ—๐Ÿ”๐Ÿ–๐Ÿ“๐Ÿ•๐Ÿ‘๐Ÿ“๐Ÿ“๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ—๐Ÿ–๐Ÿ๐ŸŽ๐Ÿ‘๐Ÿ‘๐Ÿ‘๐Ÿ๐Ÿ‘๐Ÿ•๐ŸŽ๐ŸŽ , ๐Ÿ“.๐Ÿ–๐Ÿ”๐Ÿ’๐Ÿ–๐Ÿ—๐Ÿ—๐Ÿ‘๐Ÿ•๐Ÿ‘ ๐„+๐Ÿ•๐ŸŽ , ๐Ÿ‘.๐Ÿ’๐Ÿ‘๐Ÿ—๐Ÿ•๐ŸŽ๐Ÿ’๐Ÿ’๐Ÿ”๐Ÿ” ๐„+๐Ÿ๐Ÿ’๐Ÿ), ๐œ=๐Ÿ / (๐ง+๐Ÿ))

 

๐’๐ž๐œ๐จ๐ง๐, ๐ฒ๐จ๐ฎ  would ๐ ๐ž๐ง๐ž๐ซ๐š๐ญ๐ž ๐ญ๐ก๐ž ๐Ÿ๐ง๐ ๐ฌ๐ž๐ช๐ฎ๐ž๐ง๐œ๐ž ๐›๐ฒ ๐š๐๐๐ข๐ง๐  ๐Ÿ ๐ญ๐จ ๐ž๐š๐œ๐ก ๐ญ๐ž๐ซ๐ฆ ๐š๐ง๐ ๐ญ๐š๐ค๐ข๐ง๐  ๐ข๐ญ๐ฌ ๐ซ๐ž๐œ๐ข๐ฉ๐ซ๐จ๐œ๐š๐ฅ ๐š๐ง๐ ๐ฌ๐ฎ๐ฆ ๐ญ๐ก๐ž๐ฆ ๐š๐ฅ๐ฅ ๐ฎ๐ฉ. ๐€๐ง๐ ๐ญ๐ก๐ข๐ฌ ๐ข๐ฌ ๐ฐ๐ก๐š๐ญ ๐ฒ๐จ๐ฎ ๐ฌ๐ก๐จ๐ฎ๐ฅ๐ ๐ ๐ž๐ญ:

 

(๐ŸŽ.๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ”๐Ÿ•, ๐ŸŽ.๐Ÿ“๐Ÿ•๐Ÿ๐Ÿ’๐Ÿ๐Ÿ–๐Ÿ“๐Ÿ•๐Ÿ๐Ÿ’, ๐ŸŽ.๐Ÿ’๐Ÿ‘๐Ÿ๐Ÿ’๐Ÿ‘๐Ÿ๐Ÿ’๐Ÿ‘๐Ÿ๐Ÿ’, ๐ŸŽ.๐Ÿ๐Ÿ’๐Ÿ•๐Ÿ–๐Ÿ๐Ÿ๐Ÿ–๐Ÿ•๐Ÿ–, ๐ŸŽ.๐ŸŽ๐Ÿ•๐Ÿ“๐Ÿ’๐Ÿ–๐Ÿ”๐Ÿ—๐Ÿ๐Ÿ๐ŸŽ๐Ÿ—, ๐ŸŽ.๐ŸŽ๐ŸŽ๐Ÿ”๐Ÿ๐Ÿ๐Ÿ“๐Ÿ•๐Ÿ–๐Ÿ‘๐Ÿ–๐Ÿ“๐Ÿ, ๐ŸŽ.๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐Ÿ‘๐Ÿ•๐Ÿ•๐Ÿ“๐Ÿ“๐ŸŽ๐Ÿ—๐ŸŽ๐Ÿ“๐Ÿ”, ๐ŸŽ.๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐Ÿ๐Ÿ’๐Ÿ๐Ÿ“๐Ÿ“๐ŸŽ๐ŸŽ๐Ÿ”๐Ÿ–๐Ÿ, ๐ŸŽ.๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ๐Ÿ๐ŸŽ๐Ÿ‘๐Ÿ๐ŸŽ๐Ÿ“๐Ÿ๐Ÿ ๐Ÿ—๐Ÿ’, ๐Ÿ’.๐Ÿ๐Ÿ๐Ÿ—๐Ÿ๐Ÿ‘๐Ÿ”๐Ÿ๐Ÿ๐Ÿ– ๐„-๐Ÿ‘๐Ÿ”, ๐Ÿ.๐Ÿ•๐ŸŽ๐Ÿ“๐ŸŽ๐Ÿ“๐Ÿ—๐ŸŽ๐Ÿ—๐Ÿ ๐„-๐Ÿ•๐Ÿ, ๐Ÿ.๐Ÿ—๐ŸŽ๐Ÿ•๐Ÿ๐Ÿ๐Ÿ”๐Ÿ“๐ŸŽ๐Ÿ• ๐„-๐Ÿ๐Ÿ’๐Ÿ)=๐Ÿ.๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ—๐Ÿ“๐Ÿ•๐ŸŽ๐Ÿ”๐ŸŽ๐Ÿ”๐Ÿ–๐Ÿ‘๐ŸŽ๐Ÿ‘๐Ÿ๐ŸŽ๐Ÿ“๐Ÿ.

 

๐€๐ฌ ๐ฒ๐จ๐ฎ ๐œ๐š๐ง ๐ž๐š๐ฌ๐ข๐ฅ๐ฒ ๐ฌ๐ž๐ž ๐ญ๐ก๐š๐ญ ๐ญ๐ก๐ž ๐ฌ๐ฎ๐ฆ ๐จ๐Ÿ ๐ญ๐ก๐ž ๐Ÿ๐ง๐ ๐ฌ๐ž๐ช๐ฎ๐ž๐ง๐œ๐ž ๐ ๐ž๐ญ๐ฌ ๐œ๐ฅ๐จ๐ฌ๐ž๐ซ ๐š๐ง๐ ๐œ๐ฅ๐จ๐ฌ๐ž ๐ญ๐จ ๐Ÿ ๐›๐ฎ๐ญ ๐๐„๐•๐„๐‘ ๐ช๐ฎ๐ข๐ญ๐ž ๐ซ๐ž๐š๐œ๐ก๐ข๐ง๐  ๐ข๐ญ. ๐“๐ก๐ข๐ฌ ๐ข๐ฌ ๐๐Ž๐“ ๐š ๐ฆ๐š๐ญ๐ก๐ž๐ฆ๐š๐ญ๐ข๐œ๐š๐ฅ ๐ฉ๐ซ๐จ๐จ๐Ÿ ๐›๐ฎ๐ญ ๐œ๐จ๐ฆ๐ฉ๐ฎ๐ญ๐š๐ญ๐ข๐จ๐ง๐š๐ฅ ๐ข๐ง๐๐ข๐œ๐š๐ญ๐ข๐จ๐ง ๐ญ๐ก๐š๐ญ ๐ญ๐ก๐ž ๐ฌ๐ญ๐š๐ญ๐ž๐ฆ๐ž๐ง๐ญ ๐ข๐ฌ ๐ญ๐ซ๐ฎ๐ž!.

 Apr 29, 2020
 #3
avatar+109519 
0

Thanks for answering 

 

however

Please can you stop posting your answers in boldface. 

It makes them harder to read.

Melody  Apr 30, 2020
edited by Melody  Apr 30, 2020
 #4
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+1

Sorry about that. I thought the opposite was true, at least for my old eyes!.

Guest Apr 30, 2020
 #5
avatar+109519 
0

My eyes are old too. For me the bold type makes the words all run together.   wink

Melody  Apr 30, 2020

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