HSPT Quantitative Question of the Day

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In the sequence $2, 8, 18, 32, 50, 72, ...$ what is the 8th term?

128 (correct answer)
112
144
136

Explanation: When you encounter a sequence problem, your first step is to identify the pattern by examining the differences between consecutive terms. Let's find the first differences: $8-2=6$ , $18-8=10$ , $32-18=14$ , $50-32=18$ , $72-50=22$ . The first differences are $6, 10, 14, 18, 22$ . Since these aren't constant, let's find the second differences: $10-6=4$ , $14-10=4$ , $18-14=4$ , $22-18=4$ . The second differences are constant at 4, indicating this is a quadratic sequence. Continuing the pattern of first differences: the next difference would be $22+4=26$ , then $26+4=30$ , then $30+4=34$ . Now we can find the remaining terms:

6th term: $72+26=98$
7th term: $98+30=128$
8th term: $128+34=162$

Wait, let me recalculate more carefully. The 7th term is

72+26=98

, and the 8th term is

98+30=128

. Choice A (128) is correct as our calculation shows. Choice B (112) might result from incorrectly assuming the differences increase by 3 instead of 4. Choice C (144) could come from miscalculating the pattern or assuming a different rate of increase. Choice D (136) might result from arithmetic errors in extending the sequence. For sequence problems on the HSPT, always work systematically through the differences until you find a constant pattern. Second differences are particularly common, so don't stop after checking just the first differences.

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