find the sum of all even numbers from 10 to 300, excluding those which are multiples of 8.
A 17014
B 2345
C 4562
D 9128
E none of these
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Hello vaibhav101.vaibhav101 wrote:find the sum of all even numbers from 10 to 300, excluding those which are multiples of 8.
A 17014
B 2345
C 4562
D 9128
E none of these
I would solve it as follows: $$10+12+14+\cdots+296+298+300=2\left(5+6+7+8+\cdots+148+149+150\right)$$ $$2\left(\left(1+2+3+4+5+6+7+8+\cdots+148+149+150\right)-\left(1+2+3+4\right)\right)$$ Now, the sum from 1 to 150 is equal to $$1+2+3+\cdots+150=\sum_{i=1}^{150}i=\frac{150\cdot\left(150+1\right)}{2}=75\cdot151=11325.$$ Therefore, we have $$2\left(\sum_{i=1}^{150}i\ \ \ \ -\ \left(1+2+3+4\right)\right)=2\left(11325-\left(10\right)\right)=2\left(11315\right)=22630.$$ But, this last sum includes the multiples of 8, so we have to subtract them.
First, we have to identify which are the multiples of 8:
16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, . . . , 288, 296.
Now, we have to calculate the sum of all the numbers above $$16+24+32+40+48+56+\cdots+288+296=8\left(2+3+4+5+6+7+\cdots+36+37\right)$$ $$8\left(\left(1+2+3+4+5+6+7+\cdots+36+37\right)-1\right)=8\left(\sum_{i=1}^{37}i\ \ \ \ -1\right)=8\left(\frac{37\cdot\left(37+1\right)}{2}\ \ \ -1\right)$$ $$=8\left(\frac{37\cdot38}{2}-1\right)=8\left(37\cdot19-1\right)=8\left(703-1\right)=8\cdot702=5616.$$ Hence, the sum of all even numbers from 10 to 300, excluding those which are multiples of 8 is equal to $$22630-5616=17014.$$ This tells us that the correct answer is the option A.
I hope it is clear enough.
Regards.
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For any EVENLY SPACED SET:vaibhav101 wrote:find the sum of all even numbers from 10 to 300, excluding those which are multiples of 8.
A 17014
B 2345
C 4562
D 9128
E none of these
Count = (biggest - smallest)/(increment) + 1.
Average = (biggest + smallest)/2.
Sum = (count)(average).
The INCREMENT is the difference between successive values.
Even numbers between 10 and 300, inclusive:
Here, the integers are EVEN, so the increment = 2.
Count = (300-10)/2 + 1 = 146.
Average = (300 + 10)/2 = 155.
Sum = (146)(155) = 22,630.
Multiples of 8 between 10 and 300, inclusive:
Here, the integers are MULTIPLE OF 8, so the increment = 8.
Smallest multiple of 8 = 16.
Biggest multiple of 8 = 296.
Thus:
Count = (296-16)/8 + 1 = 36.
Average = (296+16)/2= 156.
Sum = (36)(156) = 5616.
Subtracting the second sum from the first, we get:
22630 - 5616 = 17,014.
The correct answer is A.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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