Ones, Ones Everywhere

Algebra Level 2

An arithmetic sequence a 1 , a 2 , a 3 , … , a 1111 a_1, a_2, a_3, \ldots, a_{1111} with common difference d d is given. It is known that 11 a 111 = a 1111 11a_{111} = a_{1111} . Find the value of a 1111 d \frac{a_{1111}}{d} .


The answer is 1100.

Vincent Tandya by Vincent Tandya , 22, Indonesia

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1 solution

Vincent Tandya
Jan 12, 2016

Note that a 111 = a 1 + 110 d a_{111} = a_1 + 110d and a 1111 = a 1 + 1110 d a_{1111} = a_1 + 1110d .

Therefore, 11 a 111 − a 1111 = 11 ( a 1 + 110 d ) − ( a 1 + 1110 d ) = 10 a 1 + 100 d 11a_{111} - a_{1111} = 11(a_1 + 110d) - (a_1 + 1110d) = 10a_1 + 100d

However, 11 a 111 = a 1111 11a_{111} = a_{1111} . Thus, 11 a 111 − a 1111 = 0 11a_{111} - a_{1111} = 0 and so, 0 = 10 a 1 + 100 d 0 = 10a_1 + 100d . Consequently, a 1 = − 10 d a_1 = -10d .

Hence, a 1111 = a 1 + 1110 d = − 10 d + 1110 d = 1100 d a_{1111} = a_1 + 1110d = -10d + 1110d = 1100d , so a 1111 d = 1100 \frac{a_{1111}}{d} = \boxed{1100} .

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