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Difference between revisions of "2002 AMC 12P Problems/Problem 14"

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== Solution ==
 
== Solution ==
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This problem is literally almost an exact copy of [[2009 AMC 12A Problems/Problem 15|2009 AMC 12A Problem 15]], or arguably easier. Since I'm too lazy to copy over the other solutions which all work, I'll just copy the general idea of Solutions <math>1</math> and <math>2</math> from there, since that's how I solved the problem on both tests. Cycling
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Note that <math>i^4 = 1</math>, so <math>i^n = i^{4m+n}</math> for all integers <math>m</math> and <math>n</math>. In particular, <math>i = 1</math>, <math>i^2 = -1</math>, and <math>i^3 = -i</math>.
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We group the positive and negative real terms together and group the positive and negative imaginary parts together.
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The positive real terms have exponents on <math>i</math> that are multiples of 4. Therefore, the positive real part evaluates to  
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<cmath>4 + 8 + ... + 2020</cmath>
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The negative real terms have exponents on <math>i</math> that are of the form <math>4k + 2</math> for integers <math>k</math>. Therefore, the negative real part evaluates to <cmath>-(2 + 6 + ... + 2022)</cmath>
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The positive imaginary terms have exponents on <math>i</math> that are of the form <math>4k + 1</math> for integers <math>k</math>. Therefore, the negative real part evaluates to <cmath>(1 + 5 + ... + 2021)i</cmath>
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2002|ab=P|num-b=13|num-a=15}}
 
{{AMC12 box|year=2002|ab=P|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:27, 10 March 2024

Problem

Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$

$\text{(A) }-999 + 1002i \qquad \text{(B) }-1002 + 999i \qquad \text{(C) }-1001 + 1000i \qquad \text{(D) }-1002 + 1001i \qquad \text{(E) }i$

Solution

Note that $i^4 = 1$, so $i^n = i^{4m+n}$ for all integers $m$ and $n$. In particular, $i = 1$, $i^2 = -1$, and $i^3 = -i$. We group the positive and negative real terms together and group the positive and negative imaginary parts together. The positive real terms have exponents on $i$ that are multiples of 4. Therefore, the positive real part evaluates to \[4 + 8 + ... + 2020\] The negative real terms have exponents on $i$ that are of the form $4k + 2$ for integers $k$. Therefore, the negative real part evaluates to \[-(2 + 6 + ... + 2022)\] The positive imaginary terms have exponents on $i$ that are of the form $4k + 1$ for integers $k$. Therefore, the negative real part evaluates to \[(1 + 5 + ... + 2021)i\]

See also

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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
All AMC 12 Problems and Solutions

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