A community for students.
Here's the question you clicked on:
 0 viewing
Jonask
 2 years ago
A sequence of numbers (a_n) is defined as
Jonask
 2 years ago
A sequence of numbers (a_n) is defined as

This Question is Closed

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.0\[a_1=1/2\] and for each \[n \ge 2\] \[\huge a_n=\left(\frac{ 2n3 }{ 2n }\right)a_{n1}\] Prove that \[\huge\sum_{k=1}^{n}a_k<1 \] for all \[n \ge1\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1357738064697:dwJust try to Prove

Azteck
 2 years ago
Best ResponseYou've already chosen the best response.1\[a_{1}+ a_{2}+…+ a_{n} < 1\] \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2n  3 }{ 2n })a_{n1} < 1\] For n=1 LHS=\[\frac{ 1 }{ 2 }\] \[<RHS\] (Therefore) True for n=1 For n=2 LHS=\[(\frac{ 2(2)  3 }{ 2(2) })a_{21}\] \[<RHS\] (Therefore) True for n=2

Azteck
 2 years ago
Best ResponseYou've already chosen the best response.1Assume the formula is true for n=m \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2m  3 }{ 2m })a_{m1} < 1\] For n=m+1 \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2m  3 }{ 2m })a_{m1}+(\frac{ 2(m+1)  3 }{ 2(m+1) })a_{m+11} < 1\] LHS= \[1 + (\frac{ 2(m+1)  3 }{ 2(m+1) })a_{m}\] \[=(\frac{ 2m1 }{ 2m+2 })(\frac{ 2m  3 }{ 2m })a_{m1}\]

Jonask
 2 years ago
Best ResponseYou've already chosen the best response.0@sauravshakya @azteck thanks

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1I am not sure if this helps

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1359039410279:dw

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Did u mean like this @experimentX

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1yeah .. somewhat similar to that. but one terms is missing.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1there should be one n ... or i made mistake.

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1359040853679:dw

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1It was me who did a mistake

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1hmm ... now lets use this identity. http://planetmath.org/GeneratingFunctionForTheCatalanNumbers.html
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.