AMC 8 Daily Practice - The Rule of Arithmetic Sequences

Complete problem set with solutions and individual problem pages

Problem 6 Easy

Given that both sequences a_1, a_2, \ldots, a_{31} and b_1, b_2, \ldots, b_{31} are arithmetic sequences, each containing 31 terms. If a_2 + b_{30} = 29 and a_{30} + b_2 = -9, find the total sum of these two sequences.

  • A.

    29

  • B.

    60

  • C.

    200

  • D.

    310

  • E.

    400

Answer:D

For arithmetic sequences, the sum of terms equidistant from the ends is constant: a_1 + a_{31} = a_2 + a_{30}, \quad b_1 + b_{31} = b_2 + b_{30}.

By pairing terms symmetrically: (a_1 + b_{31}) + (b_1 + a_{31}) = (a_2 + b_{30}) + (b_2 + a_{30}) = 29 + (-9) = 20.

There are 15 such pairs.

Including the middle terms (which also sum to 10), the total sum is: 15 \times 20 + 10 = 310.

Final result: \boxed{310}

Related Problems
Problem 4 AMC 8 Daily Practice - The Rule of Arithmetic Sequences
Problem 5 AMC 8 Daily Practice - The Rule of Arithmetic Sequences
Problem 7 AMC 8 Daily Practice - The Rule of Arithmetic Sequences
Problem 8 AMC 8 Daily Practice - The Rule of Arithmetic Sequences
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