mun23 wrote:
In a certain sequence, each term, starting with the 3rd term, is found by multiplying the previous two terms. What is the difference between the 6th and 3rd terms in the sequence?
(1) The 1st term is equal to 8 times the 2nd term.
(2) The 4th term is equal to 1.
Manhattan OE:We have been asked for the difference between the 6th and 3rd terms of a sequence. There is little we can do to rephrase the question, but it is important to note that we don’t necessarily need to know the exact values of these terms to find their difference. We can solve as long as we know the relative values of the two terms, i.e. how much larger or smaller the 6th term is than the 3rd.
(1) INSUFFICIENT: We have been given the relationship between the 1st and 2nd terms. If we let x be the value of the 2nd term, then the 1st term is 8x. Since each term is determined by multiplying the previous two terms, we can build the whole sequence in terms of x. We can begin by multiplying 8x, the1st term, and x, the 2nd term, to get the 3rd term, 8x2. If we continue to build, we get the following sequence (starting from the 1st term): 8x, x, 8x2, 8x3, 82x5, 83x8
The difference between the 6th and 3rd terms is 83x8 − 8x2, but we can’t calculate the actual value without knowing the value of x.
(2) SUFFICIENT: At first glance, this statement does not appear to provide much helpful information. However, let’s look at what this means for the rest of the sequence. The 4th term is 1. We can find the 5th term by multiplying the 3rd and 4th terms. Let’s use n to represent the unknown 3rd term, and see what happens.
3rd term: n
4th term: 1
5th term: n*1 = n
6th term: 1*n = n
The presence of the 1 in the 4th spot produces a pair of “copycat” terms afterward! In other words, the 6th and 3rd terms are equal. Therefore, the difference between them is 0 and the correct answer is B.
Alternatively, we could approach this problem using Smart Numbers.
(1) INSUFFICIENT: We can try different numbers for the first two terms to see whether we get different results.
If we make the first two terms 8 and 1, we produce this sequence:
8, 1, 8, 8, 82, 83
What other numbers can we try? If we increase our 2nd term by 1, we get 16 and 2, but from there we will swiftly find ourselves multiplying some very large numbers! Fortunately, we are free to adjust downward and use fractions. Let’s try 4 and 1/2:
4, ½, 2, 1, 2, 2
Since 83 − 8 is clearly not the same value as 2 − 2, statement 1 is not sufficient.
(2) SUFFICIENT: While we can now use a number rather than a variable to represent the 3rd term, the result will look a lot like our previous solution:
3rd term: 4
4th term: 1
5th term: 4*1 = 4
6th term: 1*4 = 4
No matter what number we plug in, the 3rd and 6th terms must be equal, so their difference will always be 0.
The correct answer is B.