2008 UNCO Math Contest II Problems/Problem 9

Problem

Let $C_n = 1+10 +10^2 + \cdots + 10^{n-1}.$

(a) Prove that $9C_n = 10^n -1.$

(b) Prove that $(3C_3+ 2)^2 =112225.$

(c) Prove that each term in the following sequence is a perfect square: $25, 1225, 112225, 11122225, 1111222225,\ldots$

(a) $\frac{10^n-1}{9}$ (b) $112,225$ (c) $11,122,225$

2008 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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