Alternating Subscripts

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What is the value of ?

(1)

(2)

Review: Alternating Subscripts




Explanation

We've got a fine jumble of variables and subscripts in this question. We can find by finding each variable individually or by learning the value of the expression directly, perhaps through fortuitous algebra. We'll evaluate the data statements separately first, as always.

Statement (1) might have potential for fortuitous algebra, although there are other variables we probably won't be able to get rid of. To attempt to isolate , we can do



On the left side, and are both multiplied by the same thing, our expression of interest, . Factoring out that expression gives us



Without knowing what the other term is, we won't be able to determine . We can confirm by seeing different cases: the terms could equal 2 and 2, or 4 and 1. But we have a sneaky feeling this work is going to come handy. First, however, this statement is insufficient.

Statement (2) gives us the wrong expression. We want and it gives us the version with the subscripts flipped. So Statement (2) is insufficient.

Combining the statements, we can take our result from Statement (1) and substitute in the expression from Statement (2), yielding



Therefore, . We have answered the question definitively, so the statements are sufficient together.

The correct answer is (C).


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