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The sequence: $x,10x+7,10(10x+7)+7,10(10(10x+7)+7)+7,....$ $1<x<9$

Where each term, after the first, is $7$ more than $10$ times the previous term. For what value of $x$ will the $50th$ term be a multiple of $9$ ?

The answer is 8.

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The n-th term can be written as Un = 10^(n-1) x + 777...7 (n-1) digit So U50 = 10^49 (x) + 77777....7. If the number can be divided by 9, the sum of the digit must be divisible by 9. The sum of the digit is 7 (49) + x. So x=8.