Question

Fifteen integers a_1, a_2, a_3,..., a_15 are arranged in order on a number line. The integers are equally spaced and have the property that 1≤ a_1≤ 10.13≤ a_2≤ 20. and 241≤ a_15≤ 250. What is the sum of the digits of a_14 ?

(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Answer 1

To determine the sum of the digits of a_14, we calculate the common difference of the sequence and use it to find a_14, which is found to be 224. The sum of the digits 2, 2, and 4 is 8.

Step-by-step explanation:

The question asks to find the sum of the digits of a_14, the fourteenth integer in a sequence of fifteen equally spaced integers on a number line, given that a_1 is between 1 and 10, a_2 is between 13 and 20, and a_15 is between 241 and 250. To solve this problem, we need to find the common difference of the sequence and then use it to calculate a_14.

Let's assume the smallest possible values for a_1 and a_15 to find the least common difference. This gives us a_1 = 1 and a_15 = 241. The common difference d can be calculated using the formula d = (a_n - a_1) / (n - 1), which in this case is d = (241 - 1) / (15 - 1) = 240 / 14 = 17.14 (to two decimal places). Then, a_14 would be a_1 + 13d, so a_14 = 1 + 13(17.14) = 223.82, which rounds to 224 since we're dealing with integers.

The sum of the digits of 224 is 2 + 2 + 4 = 8. Therefore, the correct answer is (A) 8.