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

[size=80][b][i]Posted from my mobile device[/i][/b][/size]

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
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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
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