You are probably mixing consecutive terms in a sequence and consecutive integers: 1 and -3 are not consecutive integers, but they are consecutive terms in the sequence given. See complete solution below.

For a finite sequence of non zero numbers, the number of variations in the sign is defined as the number of pairs of consecutive terms of the sequence for which the product of the two consecutive terms is negative. What is the number of variations in sign for the sequence 1, -3, 2, 5, -4, -6?
A. 1
B. 2
C. 3
D. 4
E. 5

Given sequence: {1, -3, 2, 5, -4, -6}

The questions basically asks: how many pairs of consecutive terms are there in the sequence such that the product of these consecutive terms is negative.

1*(-3)=-3=negative;
-3*2=-6=negative;
2*5=10=positive;
5*(-4)=-20=negative;
(-4)*(-6)=24=positive.

So there are 3 pairs of consecutive terms of the sequence for which the product is negative.

Answer: C.

Hope it's clear.
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Because 1 and -4 are NOT consecutive terms in the sequence.
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Please read the question and the thread carefully. This question is answered here: for-a-finite-sequence-of-non-zero-numbers-the-number-of-127949.html#p1107497

Again, we are told that "the number of variations in sign is defined as the number of pairs of consecutive terms of the sequence ..." 1 and -3 are consecutive terms in the sequence while 5 and -6 are not.
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1. Even if we consider the terms in ascending order {-6, -4, -3, 1, 2, 5} still one pair of consecutive terms will make negative product: -3*1=-1=negative. But in this case, ANY sequence of non-zero integers which have both negative and positive numbers will have variation of 1 and the question does not make sense any more.

2. A sequence by definition is already an ordered list of terms. So if we are given the sequence of 10 numbers: 5, 6, 0, -1, -10, -10, -10, 3, 3, -100 it means that they are exactly in that order and not in another.

Hope it's clear.
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People might do a lot of things. The point is to read the stem carefully.
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The more natural understanding of the number of variations in sign is the number of times a term in the sequence has the opposite sign of its previous term. Because when the sign changes and a term and its preceding term have opposite signs, their product is necessarily negative, so the definition given is functionally equivalent. Understanding that helped me confirm that I understood what was meant by "number of variations in sign."

Keys to this problem: (1) Have in the front of your mind that the product of a positive and negative number is negative, whereas the product of two numbers of the same sign is positive, and (2) understand what number sequences are.

It seems to me that the answer is written explicitly in the question since the question says "the number of variations in sign is defined as the number of paires etc now what does it mean the number of variations? Isn't that the 3 negative signs attached the to the number?
I don't quiet get what they mean the number of variations in sign

Thank you for your response, but I still don't get why do you call 1•-3= -3 a variation. It's a variation compared to what? I can maybe understand a variation in sign when u take 2 negative numbers and multiply them u get a positive but here the negative -3 was negative already before the multiplication so what is the change that we refer as a variation?
Thanks.

(1) Tabulating the problem to see it more clearly.

For a finite sequence of non zero numbers,
the number of variations in sign is defined as the number of pairs of consecutive terms of the sequence for which the product of the two consecutive terms is negative.

What is the number of variations in sign for the sequence 1, -3, 2, 5, -4, -6 ?


(2) Converting to a more compact version.

For a finite sequence of non zero numbers,
V = number of pairs of consecutive terms in {1, -3, 2, 5, -4, -6} when product of the two consecutive terms is negative.

What is V=?

The effort is spent in figuring out what V is. Do not get confused with consecutive terms and consecutive integers. Clearly the author of the problem wants you to confuse the concept of consecutive integers and terms, however, resist the temptation.

In Set S = {a, b, c}, the terms a and b are consecutive terms, as well as b and c. However, a and c are not consecutive terms.

As such, consecutive terms for {1, -3, 2, 5, -4, -6} are:
{1, -3, 2, 5, -4, -6} = 1* -3 = -3 (negative)
{1, -3, 2, 5, -4, -6} = -3 * 2 = -6 (negative)
{1, -3, 2, 5, -4, -6} = 2* 5 = 10 (positive)
{1, -3, 2, 5, -4, -6} = 5 * -4 = -20 (negative)
{1, -3, 2, 5, -4, -6} = -4 * -6 = 24 (positive)

Note that you do not even need to multiply the numbers, however, you need to realize what happens when you multiply a negative times a positive or vice versa.

As a result of the analysis above, you can conclude that you would have three (3) negative pairs.

We are given the following sequence of numbers: 1, -3, 2, 5, -4, -6.

Every time a pair of consecutive terms product a negative product we have a “variation in sign”. We must determine how many variations in sign are in the sequence.

1 x (-3) = -3, so this is a variation in sign

(-3) x 2 = -6, so this is a variation in sign

5 x (-4) = -20, so this is a variation in sign

Thus, there is a total of 3 variations in sign.

Answer: C
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