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: \[25, 1225, 112225, 11122225, 1111222225,\ldots\]


Solution

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

See Also

2008 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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