Difference between revisions of "2020 AMC 10A Problems/Problem 8"
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| − | We can break this entire sum down into <math>4</math> integer bits, in which the sum is <math>2x</math>, where <math>x</math> is the first integer in this bit. We can find that the first sum of every sequence is <math>4x-3</math>, which we plug in for the <math>50</math> bits in the entire sequence is <math>1+2+3+\cdots+50=1275</math>, | + | We can break this entire sum down into <math>4</math> integer bits, in which the sum is <math>2x</math>, where <math>x</math> is the first integer in this bit. We can find that the first sum of every sequence is <math>4x-3</math>, which we plug in for the <math>50</math> bits in the entire sequence is <math>1+2+3+\cdots+50=1275</math>, so then we can plug it into the first term of every sequence equation we got above <math>4(1275)-3(50)=4950</math>, and so the sum of every bit is <math>2x</math>, and we only found the value of <math>x</math>, the sum of the sequence is <math>4950\cdot2=\boxed{(B)9900}</math>. |
| + | -middletonkids | ||
==See Also== | ==See Also== | ||
Revision as of 22:28, 31 January 2020
Problem
What is the value of
Solution
Split the even numbers and the odd numbers apart. If we group every 2 even numbers together and add them, we get a total of 
. Summing the odd numbers is equivalent to summing the first 100 odd numbers, which is equal to 
. Adding these two, we obtain the answer of 
.
We can break this entire sum down into 
integer bits, in which the sum is 
, where 
is the first integer in this bit. We can find that the first sum of every sequence is 
, which we plug in for the 
bits in the entire sequence is 
, so then we can plug it into the first term of every sequence equation we got above 
, and so the sum of every bit is , and we only found the value of , the sum of the sequence is 
.
-middletonkids
See Also
| 2020 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 10 Problems and Solutions | ||
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