The infinite sequence a1, a2,...,an is defined such that an

Look at the pattern:
\(an = \frac{(n + 2)}{n}\)
So a1 = 3/1
a2 = 4/2
a3 = 5/3
a4 = 6/4

Now when we multiply the first ten terms (n = 1 to 10), \(\frac{3}{1} * \frac{4}{2} * \frac{5}{3} * \frac{6}{4} * ...* \frac{11}{9} * \frac{12}{10}\)
denominators and numerators will get canceled, 3 with 3, 4 with 4 etc. Only the first two denominators and last two numerators will be left. The product will be 11*12/2 = 66

Since this is a sequence and seeing the pattern " increase of numerator and denominator by 1/1" we determine the next term.This part I understood. But fundamental thing that I have not yet grasped is how to visualise that the respective denominators and numerators actually cancel without enumerating them all. I had to list out all the fractions (just by adding 1/1 to the next) to see that they do cancel and what the final fractions would remain (though in matters of time I did not consume much time-few extra seconds). Is there a logic behind this visualisation ? Please advice.

The nth term is (n+2)/n.

So I say, when I multiply terms, they will look like this:
\(\frac{[highlight](n+2)[/highlight]}{n} * \frac{(n+3)}{(n+1)} * \frac{(n+4)}{(n+2)} * \frac{(n+5)}{(n+3)} * ... * \frac{(n+12)}{(n+10)}\)
Watch what happens here. The numerator (n+2) of the 1st term has appeared in the denominator of the 3rd term so they will cancel out. So 2nd term will have numerator that is 1 more than the numerator of the 1st term and 4th term will have a denominator that is one more than the denominator of the 3rd term and hence they will be equal and will cancel out. This will continue till we run out of denominators to cancel. So even without enumerating, we know that we will be left with first two numerators and last two denominators.

(I wish I could strike the relevant terms with different colored lines but I haven't been able to figure out how to use other operators in fractions!)

How do we know a1=1? we don't know the value of the first 10 terms? how can we assume a1=1? I didn't see that and was confused by the notation.

\(a_1\) doesn't equal to 1, it equals to \(a_1=\frac{1+2}{1}=\frac{3}{1}\).

The infinite sequence a1, a2,..., an is defined such that an = (n+2) / n for all n ≥ 1. What is the product of the first 10 terms of the sequence?
(A) 45
(B) 66
(C) 90
(D) 121
(E) 132

Stem gives the formula to get the terms of the sequences: \(a_n=\frac{n+2}{n}\) for ALL \(n\geq{1}\) (so starting from \(a_1\)). So:

\(a_1=\frac{1+2}{1}=\frac{3}{1}\);

\(a_2=\frac{2+2}{2}=\frac{4}{2}\);

\(a_3=\frac{3+2}{3}=\frac{5}{3}\);
...
\(a_{10}=\frac{10+2}{10}=\frac{12}{10}\);

The product of the first 10 terms of the sequence will equal: \(\frac{3}{1}*\frac{4}{2}*\frac{5}{3}*...*\frac{12}{10}=\frac{(3*4*5*6*6*8*9*10)*11*12}{1*2*(3*4*5*6*6*8*9*10)}=\frac{11*12}{2}=66\).

Answer: B.

Hope it's clear.

Thanks Karishma. I took a couple of bounces before I understood from your explanation albeit in own fashion (and hopefully correct!).The way I understand is this- the first two denominators (n & n+1) do not have their mirror numerators and similarly the final two numerators (n+11) & (n+12) wont get to see their mirror denominator images as the process stops at denominator (n+10) and hence dont cancel out.

This is what is given to you in the question:
The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1.

The product of first 10 terms is asked.

Using the rule, an = (n+2) / n, you can find the nth term easily.
The first term will have n = 1
a1 = (1+2)/1 = 3/1
a2 = (2+2)/2 = 4/2
and so on...

That's absolutely correct.

Seeing that the series is a sequence of fractions.
Seeing that the question is asking about the product...
we know that factors will cancel out and easily reveal the answer.

\(3 * 2 * \frac{5}{3}*\frac{3}{2}*\frac{7}{5}*\frac{8}{6}*\frac{9}{7}*\frac{10}{8}*\frac{11}{9}*\frac{12}{10}\)

After carefully cancelling out we would be left with

\(11*6 = 66\)

Answer: B

Given: The infinite sequence a1, a2,...,an is defined such that an = (n+2) / n for all n ≥ 1.
Asked: What is the product of the first 10 terms of the sequence?

The product of first 10 terms = (3/1)*(4/2)*(5/3)*(6/4)*(7/5)*(8/6)*(9/7)*(10/8)*(11/9)*(12/10) = 11*12/1*2 = 66

As other terms in numerator and denominator cancel out each other.

IMO B

Posted from my mobile device

For the given equation :
The nth term of the sequence is given by : $$\frac{\left(n+2\right)}{n}$$
The first term is : $$\frac{\left(1+2\right)}{1}=\ 3$$
The second term : $$\frac{\left(2+2\right)}{2}=\ 2$$
...............
The product of the first ten terms is represented by :
$$\left(\frac{3}{1}\right)\cdot\left(\frac{4}{2}\right)\cdot\left(\frac{5}{3}\right)\cdot.........\left(\frac{11}{9}\right)\cdot\left(\frac{12}{10}\right)$$
The numerator of 1st term is canceled with the denominator of the third term. This continues for the numerators of terms 1 to 8 and for the denominators of terms 3 to 10.
The remaining part of the product after canceling out the terms is :
$$\frac{\left(11\cdot12\right)}{\left(1\cdot2\right)}=\ \frac{132}{2}=\ 66.$$

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