In a certain the sequence, tn = t(n − 1)/t(n - 2) for n 2. If t1 = 3

Question

In a certain the sequence,  t_n=\frac{t_{n−1}}{t_{n−2}}  for  n > 2 . If  t_1=3  and  t_2=5 , then what is the product of the first 54 terms of the sequence?

A) 3/5
B) 1/3
C) 1
D) 5/3
E) 5

newyork2012 · Answer

t1=3
t2=5
t3=5/3
t4=1/3
t5=1/5
t6=3/5
Repeat again
t7=3
t8=5

54/6=9
t1*t2.....*t54=(t1*t6)^9
t1*...t6=1
So....1^9=1

So 1

Posted from my mobile device

QuantMadeEasy · Answer

Step1 : 
Finding the pattern: 
t1 = 3, t2 = 5, t3=5/3, t4 = 1/3, t5 = 1/5, t6 = 3/5
t7 = 3, t8 = 5.... so on..
After 6th term first term repeats itself
There is a cycle of 6 terms.

Step 2 : 
Product of first six terms is 1.

Step 3 : 
Finding the product of 54 terms
For 54 terms there will be 9 cycles of 6 terms each, having product of 1, total product = 1

C is correct

BrentGMATPrepNow · Answer

Given:    t_n=\frac{t_{n−1}}{t_{n−2}}

t_1=3 
 t_2=5

So,  t_3=\frac{t_{3−1}}{t_{3−2}}=\frac{t_{2}}{t_{1}}=\frac{5}{3}

t_4=\frac{t_{4−1}}{t_{4−2}}=\frac{t_{3}}{t_{2}}=\frac{(\frac{5}{3})}{5} =\frac{1}{3}

t_5=\frac{1}{5}

t_6=\frac{3}{5}

t_7=3

t_8=5 
.
.
.
As we can see, the   pattern repeats itself every 6 terms  . 
Also notice that the product of the first six terms is 1. 
This means the product of the NEXT six terms must also be 1
And the product of the six terms AFTER THAN must be 1 as well

The first 54 terms will consist of nine groups consisting of 6 terms each. 
So, the product of the first 54 terms will equal the product of nine 1's
So, the product must equal 1

Answer: C

Cheers,
Brent

Oppenheimer1945 · Answer

t1=3,t2=5,t3=5/3, t4=1/3, t5=1/5, t6=3/5, t7=3, t8=5

t1...t6=1
t7...t12=1

t49..t54=1

t1.....t54=1

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In a certain the sequence, tn = t(n − 1)/t(n - 2) for n > 2. If t1 = 3

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