Question 20

We say that a positive integer is arithmetically sequenced if its digits, in order, form an arithmetic sequence. Then the number of 4-digit positive integers which are arithmetically sequenced, is

Name: We say that a positive integer is arithmetically sequenced if its digits, in order, form an arithmetic sequence. Then the number of 4-digit positive integers which are arithmetically sequenced, is
Uploaded: 2026-08-31T17:53:50.169729+05:30
Duration: 2 min 50 s
Description: We say that a positive integer is arithmetically sequenced if its digits, in order, form an arithmetic sequence. Then the number of 4-digit positive integers which are arithmetically sequenced, is

Correct Answer: 30

Solution

Write the digits as $$a, a + d, a + 2d, a + 3d$$ with $$a \geq 1$$ and every digit between 0 and 9. For $$d = 0, 1, 2$$ the leading digit can take 9, 6 and 3 values respectively, and $$d \geq 3$$ is impossible. For $$d = -1, -2, -3$$ there are 7, 4 and 1 choices, and $$d \leq -4$$ is impossible. The total is $$9 + 6 + 3 + 7 + 4 + 1 = 30$$.

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