Four numbers are arranged in ascending order. When the 1st two numbers are increased by the number equal to their position, and the last two numbers (i.e., 3rd and 4th numbers) are increased by 2 and 4 respectively, the series thus formed is in A.P. If the 1st number is 2 less than the 2nd number, and the sum of all the numbers is 30, then find the average of the last three numbers.
ATQ, Let the numbers be a, b, c, and d. According to the question, (b + 2) – (a + 1) = (c + 2) – (b + 2) Or, 2b = a + c + 1…… (1) Also, (c + 2) – (b + 2) = (d + 4) – (c + 2) Or, 2c = b + d – 2…… (2) Let a = x Then, b = x + 2 From (1), c = x + 4 And, from (2), d = x + 6 According to the question, (x + x + 2 + x + 4 + x + 6) = 30 Or, 4x + 12 = 30 Or, 4x = 18 Or, x = 4.5 Therefore, average of last three numbers = (x + 2 + x + 4 + x + 6) / 3 = 23.5 / 3 = 7.83
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