Question 1195898
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Suppose that u_n is a sequence of real numbers satisfying u_(n+1) = 2u_(n+1) + u_n, 
and that u_3=9 and u_6=128. What is u_5 ?
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<pre>
From the basic formula

    u_(n+1) = 2u_(n+1) + u_n


you have, reducing the term u_(n+1) in both sides,

    0 = u_(n+1) + u_n,

or, EQUIVALENTLY,

    u_(n+1) = -u_n.


This formula means that the terms {u_n}, as a sequence, simply change the sign, from any given term to the next one.


Having it in mind, you may conclude that the problem is posed INCORRECTLY, 

since u_6 must be -9 then.
</pre>

This notice completes the solution.



<U>ANSWER</U>.  &nbsp;&nbsp;The problem is posed &nbsp;INCORRECTLY: &nbsp;it is &nbsp;SELF-CONTRADICTORY.



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<U>comment from student</U>: Ooops! I revised it here:https://artofproblemsolving.com/texer/japjtbbh



<U>My response</U> :  &nbsp;&nbsp;In such cases, &nbsp;you should apology and re-submit your problem to the forum, &nbsp;again.


I will solve the updated version below.



///////// - - - the updated version - - - \\\\\\\\\\\



Suppose that u_n is a sequence of real numbers satisfying u_(n+2) = 2u_(n+1)+u_n, 
and that u_3=9 and u_6=128. What is u_5?
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<pre>
Let x be u_4.


My goal is to find x from given data.

Then I will be in position to find u_5.


So, for u_5 I have then, using the basic formula

    u_5 = 2u_4 + u_3 = 2x + 9.


For u_6 I will have

    u_6 = 2*u_5 + u_4 = 2*(2x+9) + x = (4x+18) + x = 5x + 18.


From the other side hand,  u_6 = 128  (given).

It gives me an equation

    5x + 18 = 128

or

    5x      = 128-18 = 110,

     x                 110/5 = 22.


Now  u_5 = 2u_4 + u_3 = 2x + 9 = 2*22 + 9 = 44 + 9 = 53.   


<U>ANSWER</U>.  u_5 = 53.
</pre>

Solved.