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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, . . ., sn have an even number of sign changes? – OG 2020 Question #312 with Solution

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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, . . ., sn have an even number of sign changes?

  1. sk = (–1)k for all positive integers k from 1 to n.
  2. n is odd.
Source OG 2020
Type Data Sufficiency
Topic Algebra
Sub-Topic Sequences
Difficulty Medium – 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, …, sn have an even number of sign changes or not.

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

Step 3: Analyse Statement 1

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

  • It depends on the value of k – whether the number of sign changes will be even or odd.

As we don’t know the exact value of k, 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.

Step 5: Combine Both Statements Together (If Needed)

Combining both statements, we can say

  • sk = (–1)k for all positive integers k from 1 to n.
  • Also, n is odd.
    • Therefore, there will be even number of sign changes
    • For example, if n = 3, then the numbers are -1, 1, -1 (2 sign changes)
    • Similarly, if n = 5, then the numbers are -1, 1, -1, 1, -1 (4 sign changes)

As we can determine 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.

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