Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 5"
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− | Suppose that <math>f</math> is a function with the property that for all <math>x</math> and <math>y, f(x + y) = f(x) + f(y) + 1</math> and <math>f(1) = 2.</math> What is the value of <math>f(3)</math>? | + | Suppose that <math>f</math> is a function with the property that for all <math>x</math> and <math>y</math>, <math>f(x + y) = f(x) + f(y) + 1</math> and <math>f(1) = 2.</math> What is the value of <math>f(3)</math>? |
<center><math> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8 </math></center> | <center><math> \mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8 </math></center> | ||
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Notice that <math>f(3)=f(2+1)=f(2)+f(1)+1=f(2)+3</math>. Also, <math>f(2)=f(1+1)=f(1)+f(1)+1=5</math>. Thus, <math>f(3)=3+5=8</math>. | Notice that <math>f(3)=f(2+1)=f(2)+f(1)+1=f(2)+3</math>. Also, <math>f(2)=f(1+1)=f(1)+f(1)+1=5</math>. Thus, <math>f(3)=3+5=8</math>. | ||
− | == | + | In general, <math>f(x + 1) = f(x) + f(1) + 1 = f(x) + 3</math>, so we have a simple [[recursion|recursive]] definition for the [[function]] <math>f</math>. From here we can see that <math>f(n) = 3n - 1</math> for all [[positive integer]]s <math>n</math>. |
− | * [[University of South Carolina High School Math Contest/1993 Exam]] | + | |
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+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 4|Previous Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 6|Next Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]] | ||
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+ | [[Category:Introductory Algebra Problems]] |
Latest revision as of 14:55, 29 July 2006
Problem
Suppose that is a function with the property that for all and , and What is the value of ?
Solution
Notice that . Also, . Thus, .
In general, , so we have a simple recursive definition for the function . From here we can see that for all positive integers .