1964 AMC 12 Problem 34

Attempt Problem 34 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

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34.

If

n

is a multiple of

4,

the sum

\begin{aligned} s&=1+2i+3i^2+\cdots\\ &\quad+(n+1)i^n, \end{aligned}

where

i=\sqrt{-1},

equals:

1+i

\dfrac12(n+2)

\dfrac12(n+2-ni)

\dfrac12\left[(n+1)(1-i)+2\right]

\dfrac18(n^2+8-4ni)

Answer: C

Concepts:complex number roots of unity summation

Difficulty rating: 1870

Small hint:

Group the terms in blocks of four powers of

i

Big hint:

Each complete block beginning with coefficient

4j+1

sums to

-2-2i

Solution:

Write

n=4m.

For

j=0,\ldots,m-1,

the four terms with exponents

4j

through

4j+3

sum to

\begin{aligned} &(4j+1)+(4j+2)i\\ &\quad-(4j+3)-(4j+4)i\\ &=-2-2i. \end{aligned}

The final term is

4m+1.

Therefore

\begin{aligned} s&=m(-2-2i)+(4m+1)\\ &=\frac{n+2-ni}{2}. \end{aligned}

Thus, the correct answer is C.