Official Guide Explanation:
Problem Solving #228




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Solution and Metadata

Question: 228
Page: 185
Difficulty: 7 (Very Difficult)
Category 1: Algebra > Functions > Sequences
Category 2: Algebra > Exponents >

Explanation: Another way to phrase the definition of "arithmetic sequence" is to say that the numbers in such a sequence are equally spaced. For instance, they could be consecutive integers, consecutive evens, or consecutive multiples of 7. To look at each of the statements individually:

I.    By doubling each number, you double the distance between them, but the distance between each one is still the same. For instance, if a sequence is {1, 2, 3}, doubling each number results in {2, 4, 6}, still equally spaced.

II.    Subtracting 3 from each number may change each number, but it doesn't change the amount of space between them. For instance, the sequence {4, 5, 6} becomes {1, 2, 3}.

III.    In this case, the numbers change, and the distance between them changes as well. Instead of a consistent amount between numbers, the distance grows with each number. for example, the sequence {2, 3, 4} would become {4, 9, 16}, with differences of 5 and 7 between them--not a constant.

Thus, I and II are arithmetic sequences, while III is not. That makes (D) the correct answer.

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