ShmoopTube

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Here’s your shmoop du jour, brought to you by Patterns.

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Because no self-respecting clown ever wore a solid color tie.

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The first term in a sequence is x. Each following term is 2 less than half of the previous term.

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If x is greater than 12, what percent of the first term is the third term?

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And here are the potential answers…

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We can solve this problem while keeping x as a variable.

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The problem tells us that the following term is 2 less than half of the previous, so to find

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the second term, we subtract 2 from half of x, which is our very first term.

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Then, we repeat this process to find the next term.

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We get N sub 3 equals x over 2 minus 2 all over 2 minus 2.

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First, instead of dividing by 2, we can multiply by one-half.

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The x over 2 minus 2 still bothers us though…so let’s multiply the fraction and negative 2 by 2…

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and put them under a common denominator of 2.

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x minus 2 times 2 is 4… all over 2.

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Multiplying that by 1/2, we have x minus 4 all over 4.

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Separating it out from a common denominator, we have x over 4 minus 1.

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But…let’s not forget the minus 2 from before. Negative 1 minus 2 is negative 3.

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So n sub 3 equals x over 4 minus 3.

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Now that our equation looks much better, we can solve the problem with a little logic.

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When we divide a number by 4, we get 25% of the original number.

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A quarter is…25%.

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Now…any x over 4 MINUS 3 will make the number even smaller than 25%.

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And the only answer choice that has a percentage less than 25% is A, at 12.5%.

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Another way to solve this is by plugging in a real number.

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We can start with 24 as x, and divide by 2. We’ll get 12. Then, we subtract 2 to get 10.

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This is our second number.

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To find the third, we repeat. Divide by 2, that's 5, subtract 2, that's 3.

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3 is 12.5% of 24 and our answer is still (A).

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