The average (arithmetic mean) of the multiples of 6 that are greater than 0 and less than 1,000 is
499
500
501
502
503
answer is 501. easy solution anybody? thanks.
sounds so easy, yet so disturbing...
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number of number divisible by 6=10006/6= 165
average=sum/165
sum= 165/2(6+996)
=165*501
average=165*501/165= 501
average=sum/165
sum= 165/2(6+996)
=165*501
average=165*501/165= 501
The powers of two are bloody impolite!!

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don't know if this is the fastest way but...
sum of multiples of 6 until 1000 = 1*6 + 2*6 + 3*6... + 166*6 (which is 966, the largest multiple smaller than 1000)
so that = 6*(1+2+3+...+166)
so the average = 6*(1+2+3+...166) / 166
average of a series is equal to the median of the series.
median of series 1+2+3...166 = (83+84)/2 which is = 83.5
so final average = 6*83.5 = 501
edit: Ok, tohellandback's way is a lot faster, lol
according to tohellandback's method, the short cut is just to simply do: largest multiple + 6 then divide by 2: (996+6)/2 = 501. this is ofcourse after fact, but is there a logical reason for this?
sum of multiples of 6 until 1000 = 1*6 + 2*6 + 3*6... + 166*6 (which is 966, the largest multiple smaller than 1000)
so that = 6*(1+2+3+...+166)
so the average = 6*(1+2+3+...166) / 166
average of a series is equal to the median of the series.
median of series 1+2+3...166 = (83+84)/2 which is = 83.5
so final average = 6*83.5 = 501
edit: Ok, tohellandback's way is a lot faster, lol
according to tohellandback's method, the short cut is just to simply do: largest multiple + 6 then divide by 2: (996+6)/2 = 501. this is ofcourse after fact, but is there a logical reason for this?

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oh let me rewrite..may be i got you confused..godemol wrote:tohellandback,
please explain the part where,
sum= 165/2(6+996)
thanks...
it is actually..165/2 * (6+996)
sum of n terms in AP is n/2* ( first term + last term)
The powers of two are bloody impolite!!

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well thats not exactly what i have done..capnx wrote:don't know if this is the fastest way but...
sum of multiples of 6 until 1000 = 1*6 + 2*6 + 3*6... + 166*6 (which is 966, the largest multiple smaller than 1000)
so that = 6*(1+2+3+...+166)
so the average = 6*(1+2+3+...166) / 166
average of a series is equal to the median of the series.
median of series 1+2+3...166 = (83+84)/2 which is = 83.5
so final average = 6*83.5 = 501
edit: Ok, tohellandback's way is a lot faster, lol
according to tohellandback's method, the short cut is just to simply do: largest multiple + 6 then divide by 2: (996+6)/2 = 501. this is ofcourse after fact, but is there a logical reason for this?
sum of n terms in AP is n/2 * (1st term + last term)
thats why sum is 165/2 * (6+996)
and that gives you
165 * 501
when you take the average you divide by the total numbers of terms i.e. 165
The powers of two are bloody impolite!!

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sum of n terms is n/2 * (2a +(n1)d)godemol wrote:sum of n terms in AP is n/2 * (1st term + last term)
thanks for ur quick response...one more thing, AP=average problems? also, does this formula apply for all sum of series? thanks.
= n/2 * (a + a+ (n1)d)
a is the first terms
a+(n1)d is nothing but the last term
and of course it is applicable to all series in arithmetic progression
The powers of two are bloody impolite!!
This is the solution of this problem:
Formula for sum for arithmetic sequence:
S= N/2(A1+An)
N number of terms in the sequence.
A1 1st term
An last term.
So problem asks a multiple of six greater than 0 and smaller than 1000.
Such will be 6(1st term),12,18...966(last term)
Number of terms 166
Now just apply the formula for sequence and you have:
S= 166/2(6+966)= 83*1002= 13 778
Now, you need to find the mean. Since the sum= 13778 and number of terms= 166; Mean=13778/166=501
Make sure that you memorize this formula. It is being tested quite frequently on the actual test.
Formula for sum for arithmetic sequence:
S= N/2(A1+An)
N number of terms in the sequence.
A1 1st term
An last term.
So problem asks a multiple of six greater than 0 and smaller than 1000.
Such will be 6(1st term),12,18...966(last term)
Number of terms 166
Now just apply the formula for sequence and you have:
S= 166/2(6+966)= 83*1002= 13 778
Now, you need to find the mean. Since the sum= 13778 and number of terms= 166; Mean=13778/166=501
Make sure that you memorize this formula. It is being tested quite frequently on the actual test.

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Well, actually, in the options, the only one divisible by 6 is 501. But HellandBack method applies to any bizarre case in GMAT exam, thanks Hell.eustudent wrote:the only best short cut i caould think of is this:
Sum= 166/2(6+966)/166(number of terms in order to find the mean)
this leaves us with 1002/2= 501