|
Question 26590: Let (Tn) = (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .) be the Fibonacci sequence defined by
T1 = T2 = 1, Tn = T(n−1) + T(n−2) if n > 2.
Show that the following hold for n that is greator or equal to 1.
T2+T3+...+T(2n-1)=T(2n)
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Let (Tn) = (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .) be the Fibonacci sequence defined by
T1 = T2 = 1, Tn = T(n−1) + T(n−2) if n > 2.
Show that the following hold for n that is greator or equal to 1.
T2+T3+...+T(2n-1)=T(2n)
WE HAVE
T1=1
T2=1.......OR.......................................T2-T1=0
T3=T2+T1............................................T3-T2=T1
T4=T3+T2............................................T4-T3=T2
T5=T4+T3............................................T5-T4=T3
...........................................................................
.............................................................................
T2N=T(2N-1)+T(2N-2)...........................T2N-T(2N-1)=T(2N-2)
ADDING ALL ABOVE ,WE FIND ON THE LHS ,ALL EXCEPT T2N AND -T1 CANCEL EACH OTHER.SO WE GET
T2N-T1=T1+T2+T3+.........+T(2N-2)...OR......
T2N=T1+{T1+T2+T3+........T(2N-2)}
T2N=1+{T1+T2+T3+........T(2N-2)}...THIS IS THE ANSWER.YOUR QUESTION IS NOT CORRECT YOU CAN CHECK AS BELOW...WE HAVE
T1=1..= 1
T2=1..= 1
T3=1+1= 2
T4=2+1= 3
T5=3+2= 5
T6=5+3= 8
T7=8+5= 13
T8=13+8=21 ..SO T2N=T8=1+T1+T2+T3+T4+T5+T6=1+(1+1+2+3+5+8)=21..THAT IS
.........................T8=1+{T1+T2+T3+........T(8-2)}...AND NOT
T2N=T2+T3+...+T(2n-1) WHICH GIVES US T8=T2+T3+T4+T5+T6+T(8-1)
=1+2+3+5+8+13=32...WHICH IS NOT CORRECT AS T8=21
FURTHER FIBONACCI SEQUENCE IS
0,1,1,2,3,5,8......ETC...WITH THE PROPERTY THAT TN=T(N-1)+T(N-2)...AND T1=0 AND T2=1.
|
|
|
| |
