What is the value of the following expression? 2(1-(1)/(2)) + 3(1-(1)/(3)) + \beginaligned &4(1-(1)/(4)) + \cdots \ &\quad + 10(1-(1)/(10)) \endaligned

1998 AMC 8 Problem 12 — Solution

1998 AMC 8 Problem 12

Attempt Problem 12 of the 1998 AMC 8 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 1998 AMC 8 solutions, or check the answer key.

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12.

What is the value of the following expression? $2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) +$ $\begin{aligned} &4\left(1-\dfrac{1}{4}\right) + \cdots \\ &\quad {}+ 10\left(1-\dfrac{1}{10}\right) \end{aligned}$

$45$

$49$

$50$

$54$

$55$

Answer: A

Concepts:summation algebraic manipulation

Difficulty rating: 1090

Solution:

For each integer $n$ from $2$ through $10$ ,

$n\left(1-\frac1n\right)=n-1.$

The expression is therefore $1+2+\cdots+9=45$ .

Thus, the correct answer is A .

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