2020 AMC 10A Problems/Problem 8
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
Another solution involves adding everything and subtracting out what is not needed. The first step involves solving . To do this, we can simply multiply
and
and divide by
. This will get us
. The next step involves subtracting out the numbers with minus signs. We actually have to do this twice, because we need to take the numbers we weren’t supposed to add out and subtract them from the problem. Doing a quick calculation, we can see that from
to 200
4
50
50
51
2
4
5,100
10,200
20100-10200
\boxed{\text{(B) }9900}
Phineas1500$
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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