Difference between revisions of "2018 AMC 8 Problems/Problem 2"
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| − | ==Problem | + | ==Problem== |
| − | What is the value of the product<cmath>\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?</cmath> | + | What is the value of the product |
| + | |||
| + | <cmath>\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?</cmath> | ||
<math>\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math> | <math>\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math> | ||
==Solution== | ==Solution== | ||
| − | By adding up the numbers in each parentheses, we | + | By adding up the numbers in each of the <math>6</math> parentheses, we get: |
| + | |||
| + | <math>\frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \frac{7}{6}</math>. | ||
| + | |||
| + | Using telescoping, most of the terms cancel out diagonally. We are left with <math>\frac{7}{1}</math> which is equivalent to <math>7</math>. Thus, the answer would be <math>\boxed{\textbf{(D) }7}</math>. | ||
| + | |||
| + | == Video Solution (CRITICAL THINKING!!!)== | ||
| + | https://youtu.be/OFsrMjvR950 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/OeaCROQV4j4 | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution by OmegaLearn== | ||
| + | https://youtu.be/TkZvMa30Juo?t=3213 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| − | + | ==See also== | |
| + | {{AMC8 box|year=2018|num-b=1|num-a=3}} | ||
| − | + | {{MAA Notice}} | |
Latest revision as of 20:19, 29 March 2023
Contents
Problem
What is the value of the product
![\[\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?\]](http://latex.artofproblemsolving.com/2/c/7/2c7aacf4c691c580906f1aaa847be38a7dbf6f26.png)
Solution
By adding up the numbers in each of the 
parentheses, we get:
.
Using telescoping, most of the terms cancel out diagonally. We are left with 
which is equivalent to 
. Thus, the answer would be 
.
Video Solution (CRITICAL THINKING!!!)
~Education, the Study of Everything
Video Solution
~savannahsolver
Video Solution by OmegaLearn
https://youtu.be/TkZvMa30Juo?t=3213
~ pi_is_3.14
See also
| 2018 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
