2021 AMC 12A Problems/Problem 14
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2 (Detailed Explanation of Solution 1)
- 4 Solution 3
- 5 Solution 4 (Estimations and Answer Choices)
- 6 Video Solution by Punxsutawney Phil
- 7 Video Solution by Hawk Math
- 8 Video Solution by OmegaLearn (Using Logarithmic Manipulations)
- 9 Video Solution by TheBeautyofMath (using Magical Ability)
- 10 Video Solution by The Power of Logic
- 11 See also
Problem
What is the value of
Solution 1
This equals ~JHawk0224
Solution 2 (Detailed Explanation of Solution 1)
We will apply the following property of logarithms: which can be proven by the Change of Base Formula: Now, we simplify the expressions inside the summations: and Using these results, we evaluate the original expression: ~MRENTHUSIASM
Solution 3
First, we can get rid of the exponents using properties of logarithms:
(Leaving the single in the exponent will come in handy later). Similarly,
Then, evaluating the first few terms in each parentheses, we can find the simplified expanded forms of each sum using the additive property of logarithms:
To evaluate the exponent of the in the first logarithm, we use the triangular numbers equation:
Finally, multiplying the two logarithms together, we can use the chain rule property of logarithms to simplify:
Thus,
-Solution by Joeya
Solution 4 (Estimations and Answer Choices)
In note that the addends are greater than for all
In note that the addends are greater than for all
We have the inequality which eliminates choices and We get the answer by either an educated guess or a continued approximation:
Since it follows that By an extremely rough underestimation, From here, it should be safe to guess that the answer is
As an extra guaranty, note that Therefore, we must have
~MRENTHUSIASM
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=FD9BE7hpRvg&t=322s
Video Solution by Hawk Math
https://www.youtube.com/watch?v=AjQARBvdZ20
Video Solution by OmegaLearn (Using Logarithmic Manipulations)
Video Solution by TheBeautyofMath (using Magical Ability)
https://youtu.be/ySWSHyY9TwI?t=999
~IceMatrix
Video Solution by The Power of Logic
See also
2021 AMC 12A (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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