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PQID: DS57502.01 | OG 2020: Question No. 312

In any sequence of n nonzero numbers, a pair of consecutive terms with opposite signs represents a sign change. For example, the sequence –2, 3, –4, 5 has three sign changes. Does the sequence of nonzero numbers s₁, s₂, s₃, . . ., s_n have an even number of sign changes?

1. s_k = (–1)^k for all positive integers k from 1 to n.
2. n is odd.

Source	OG 2020
PQID	DS57502.01
Type	Data Sufficiency
Topic	Algebra
Sub-Topic	Sequences
Difficulty	Hard

Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are given

• In any sequence of n nonzero numbers, a pair of consecutive terms with opposite signs represents a sign change.
  • For example, the sequence -2, 3, -4, 5 has three sign changes.
    • First time from -2 to 3
    • Second time from 3 to -4
    • Third time from -4 to 5.

We need to determine

• Whether the sequence of nonzero numbers s₁, s₂, s₃, …, s_n has an even number of sign changes or not.

As we have no relevant information about the terms present in the sequence, let us now analyse the individual statements.

Step 3: Analyse Statement 1

As per the information given in statement 1, s_k = (–1)^k for all positive integers k from 1 to n.

So, s₁ = (-1)¹ = -1; s₂ = (-1)² = 1; s₃ = (-1)³ = -1; …, and so on.

Finally, s_n = (-1)ⁿ

• So, here the sequence becomes -1, 1, -1, …, (-1)ⁿ and has (n – 1) sign changes in total.
  • Example 1: -1, 1, -1, 1 has 4 terms and 3 sign changes.
  • Example 2: -1, 1, -1, 1, -1 has 5 terms and 4 sign changes.
• Now, everything depends on the value of n
  • If n is odd, the number of sign changes = n – 1 = even
  • If n is even, the number of sign changes = n – 1 = odd.

As we don’t know the exact value of n, we can say statement 1 is not sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 2, n is odd.

• This statement only tells us that number of elements in the series is odd.
• However, from this statement, we cannot determine the number of sign changes that happened within the elements.

Hence, statement 2 is not sufficient to answer the question.

NOTE: Never drag any information from statement 1 while analyzing statement 2.

Step 5: Combine Both Statements Together (If Needed)

Combining both statements, we can say

• s_k = (–1)^k for all positive integers k from 1 to n.
• Also, n is odd.
  • Hence, number of sign changes = n – 1 = even.

As we can determine with certainty that number of sign changes will be even, we can say that the combination of statements is sufficient to answer the question.

Hence, the correct answer choice is option C.

Takeaways:

1. To best understand any sequence given its general term, always write the first few terms. This will help in identifying a pattern that will make further processing easy.
2. In this question, once we wrote the first few terms of the sequence using the general formula given in statement 1, we could infer that the total number of sign changes was n – 1. This made further processing very easy. (Combined analysis became a one-liner!)

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