Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 9"
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+ | (a) We know that <math>C_n</math> is a geometric series, so we can define it explicitly as follows | ||
+ | |||
+ | <math>C_n=\frac{10^n-1}{9}</math> | ||
+ | |||
+ | multiplying both sides by 9 yields our answer. | ||
+ | |||
+ | |||
+ | (b) We have | ||
+ | |||
+ | <math>(3*111+2)^2=335^2</math>, | ||
+ | |||
+ | yielding <math>112,225</math>. | ||
+ | |||
+ | |||
+ | (c) We say that the nth member of the sequence equals <math>(3*C_n+2)^2</math>. Expanding yields | ||
+ | |||
+ | <math>(3*\frac{10^n-1}{9}+2)^2</math>, | ||
+ | |||
+ | <math>=(\frac{10^n+5}{3})^2</math>, | ||
+ | |||
+ | <math>=\frac{10^{2n}}{9}+\frac{10^{n+1}}{9}+\frac{25}{9}</math>. | ||
+ | |||
+ | Dividing each term separately, we know that the first term will add <math>2n</math> <math>1</math>s and <math>\frac{1}{9}</math>, the second term will add <math>n+1</math> <math>1</math>s and <math>\frac{1}{9}</math>, and the third will add <math>\frac{25}{9}</math>, giving | ||
+ | |||
+ | <math>\overbrace{11...11}^{2n}+\overbrace{11...11}^{n+1}+\frac{27}{9}</math>, | ||
+ | |||
+ | <math>\overbrace{11...11}^{n-1}\overbrace{22...22}^{n}5</math>, | ||
+ | |||
+ | which is exactly what we wanted. | ||
+ | |||
+ | |||
(a) <math>\frac{10^n-1}{9}</math> (b) <math>112,225</math> (c) <math>11,122,225</math> | (a) <math>\frac{10^n-1}{9}</math> (b) <math>112,225</math> (c) <math>11,122,225</math> | ||
Revision as of 20:48, 17 June 2021
Problem
Let
(a) Prove that
(b) Prove that
(c) Prove that each term in the following sequence is a perfect square:
Solution
(a) We know that is a geometric series, so we can define it explicitly as follows
multiplying both sides by 9 yields our answer.
(b) We have
,
yielding .
(c) We say that the nth member of the sequence equals . Expanding yields
,
,
.
Dividing each term separately, we know that the first term will add s and , the second term will add s and , and the third will add , giving
,
,
which is exactly what we wanted.
(a) (b) (c)
See Also
2008 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
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 |