AMC 8 Daily Practice - Calculation Tricks by Grouping

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Problem 8 Medium

What is the value of 32\times 31-31\times 30+30\times 29-29\times 28+…+4\times 3-3\times 2+2\times 1?

  • A.

    992

  • B.

    0

  • C.

    1

  • D.

    257

  • E.

    512

Answer:E

By observing the first four terms, we notice the common factor 31.

Applying the associative property of multiplication, the first four terms can be rewritten as: 31 \times (32 - 30)

Similarly, analyzing terms five to eight, we identify the common factor 29 and rewrite them as: 29 \times (30 - 28)

The original expression can thus be transformed into a series of such grouped terms: 32 \times 31 - 31 \times 30 + 30 \times 29 - 29 \times 28 + \dots + 4 \times 3 - 3 \times 2 + 2 \times 1 = 31 \times (32 - 30) + 29 \times (30 - 28) + \dots + 3 \times (4 - 2) + 2 \times 1 = (31 + 29 + \dots + 3) \times 2 + 2

This forms an arithmetic sequence with: - First term a_1 = 3 - Last term a_n = 31 - Common difference d = 2 - Number of terms n = \frac{31 - 3}{2} + 1 = 15

Applying Gauss's formula for the sum of an arithmetic series: S = \frac{n}{2} \times (a_1 + a_n) S = \frac{15}{2} \times (3 + 31) = 255

Final calculation: 255 \times 2 + 2 = 510 + 2 = 512

The final result is \boxed{512}.

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