Difference between revisions of "1999 AHSME Problems/Problem 20"
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== Problem == | == Problem == | ||
− | The sequence <math>a_{1},a_{2},a_{3},\ldots</math> | + | The sequence <math>a_{1},a_{2},a_{3},\ldots</math> satisfies <math>a_{1} = 19,a_{9} = 99</math>, and, for all <math>n\geq 3</math>, <math>a_{n}</math> is the arithmetic mean of the first <math>n - 1</math> terms. Find <math>a_2</math>. |
− | == Solution == | + | <math>\textrm{(A)} \ 29 \qquad \textrm{(B)} \ 59 \qquad \textrm{(C)} \ 79 \qquad \textrm{(D)} \ 99 \qquad \textrm{(E)} \ 179</math> |
+ | |||
+ | == Solution 1== | ||
Let <math>m</math> be the arithmetic mean of <math>a_1</math> and <math>a_2</math>. We can then write <math>a_1=m-x</math> and <math>a_2=m+x</math> for some <math>x</math>. | Let <math>m</math> be the arithmetic mean of <math>a_1</math> and <math>a_2</math>. We can then write <math>a_1=m-x</math> and <math>a_2=m+x</math> for some <math>x</math>. | ||
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Realizing this, one can easily prove by induction that <math>\forall n\geq 3;~ a_n=m</math>. | Realizing this, one can easily prove by induction that <math>\forall n\geq 3;~ a_n=m</math>. | ||
− | It follows that <math>m=a_9=99</math>. From <math>19=a_1=m-x</math> we get that <math>x=80</math>. And thus <math>a_2 = m+x = \ (E) boxed{179}</math> | + | It follows that <math>m=a_9=99</math>. From <math>19=a_1=m-x</math> we get that <math>x=80</math>. And thus <math>a_2 = m+x = \boxed{(E) 179}</math>. |
+ | |||
+ | == Solution 2== | ||
+ | Let <math>a_1=a</math> and <math>a_2=b</math>. Then, <math>a_3=\frac{a+b}{2}</math>, <math>a_4=\frac{a+b+\frac{a+b}{2}}{3}=\frac{a+b}{2},</math> and so on. | ||
+ | |||
+ | Since <math>a_3=a_4</math>, <math>a_n=a_3</math> for all <math>n\geq3.</math> | ||
+ | |||
+ | Hence, <math>a_9=\frac{a_1+a_2}{2}=\frac{a+b}{2}=99, a+b=198.</math> We also know that <math>a_1=a=19.</math> | ||
+ | |||
+ | Subtracting <math>a_1</math> from <math>198,</math> we get <math>b=a_2=\boxed{(E) 179}.</math> | ||
+ | |||
+ | ~Benedict T (countmath1) | ||
== See also == | == See also == |
Latest revision as of 13:00, 21 March 2023
Contents
Problem
The sequence satisfies , and, for all , is the arithmetic mean of the first terms. Find .
Solution 1
Let be the arithmetic mean of and . We can then write and for some .
By definition, .
Next, is the mean of , and , which is again .
Realizing this, one can easily prove by induction that .
It follows that . From we get that . And thus .
Solution 2
Let and . Then, , and so on.
Since , for all
Hence, We also know that
Subtracting from we get
~Benedict T (countmath1)
See also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.