2009 AMC 12A Problems/Problem 15

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For what value of $n$ is $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$?

Note: here $i = \sqrt { - 1}$.

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 97 \qquad \textbf{(E)}\ 98$

Solution 1

We know that $i^x$ cycles every $4$ powers so we group the sum in $4$s. \[i+2i^2+3i^3+4i^4=2-2i\] \[5i^5+6i^6+7i^7+8i^8=2-2i\]

We can postulate that every group of $4$ is equal to $2-2i$. For 24 groups we thus, get $48-48i$ as our sum. We know the solution must lie near The next term is the $24*4+1=97$th term. This term is equal to $97i$ (first in a group of $4$ so $i^{97}=i$) and our sum is now $48+49i$ so $n=97\Rightarrow\boxed{\mathbf{D}}$ is our answer

Solution 2

Obviously, even powers of $i$ are real and odd powers of $i$ are imaginary. Hence the real part of the sum is $2i^2 + 4i^4 + 6i^6 + \ldots$, and the imaginary part is $i + 3i^3 + 5i^5 + \cdots$.

Let's take a look at the real part first. We have $i^2=-1$, hence the real part simplifies to $-2+4-6+8-10+\cdots$. If there were an odd number of terms, we could pair them as follows: $-2 + (4-6) + (8-10) + \cdots$, hence the result would be negative. As we need the real part to be $48$, we must have an even number of terms. If we have an even number of terms, we can pair them as $(-2+4) + (-6+8) + \cdots$. Each parenthesis is equal to $2$, thus there are $24$ of them, and the last value used is $96$. This happens for $n=96$ and $n=97$. As $n=96$ is not present as an option, we may conclude that the answer is $\boxed{97}$.

In a complete solution, we should now verify which of $n=96$ and $n=97$ will give us the correct imaginary part.

We can rewrite the imaginary part as follows: $i + 3i^3 + 5i^5 + \cdots = i(1 + 3i^2 + 5i^4 + \cdots) = i(1 - 3 + 5 - \cdots)$. We need to obtain $(1 - 3 + 5 - \cdots) = 49$. Once again we can repeat the same reasoning: If the number of terms were even, the left hand side would be negative, thus the number of terms is odd. The left hand side can then be rewritten as $1 + (-3+5) + (-7+9) + \cdots$. We need $24$ parentheses, therefore the last value used is $97$. This happens when $n=97$ or $n=98$, and we are done.

Solution 3 (Fast)

Some may know the equation:


(For those curious, this comes from differentiating the equation for finite geometric sums)

Using this equation, we have

\[48+49i=i\frac{1-(n+1)i^n+ni^{n+1}}{(1-i)^2}\] \[=\frac{1-(n+1)i^n+ni^{n+1}}{-2}\] \[=-\frac{1}{2}+\frac{(n+1)i^n}{2}-\frac{ni^{n+1}}{2}\]

Since the imaginary and the real part must be positive, we know that $i^{n+1}=-1$ or $i^{n+1}=-i$. By the same line of reason, $i^{n}=1,i$. This only works for $n\equiv 1 \mod 4$. Therefore, we have:


Solving either the real or imaginary part gives $\boxed{\mathbf{(D) }97}$

Video Solution



See Also

2009 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 12 Problems and Solutions

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