GMAT prep num props

[vinviper1]

Thanks!

[ektamatta]

see...u don't have to get confused by sum of first 50 positive integers

Question tell us to find even positive integer from 102 to 200, inclusive]ve
this how we solve ..there are 50 even number between 102 to 200 including 102 & 200

this u what u need to memorize total number /2 50/2=25

then a=102 , b=200 a+b=102+200=302

now, multiply 302 * 25 u will get 7,550 answer.

[ektamatta]

hey! vinviper1

try OG 11 problem No.204. You will feel more comfortable with the formula applied in this problem.

[Ian Stewart]

You can certainly ignore the information given in the question about the sum of the first 50 positive evens, and apply a formula directly to get an answer. It's quite quick as well, so not a bad approach. Still, you can get the answer without the formula, by using the information given:

2+4+6+...+98+100 = 2550

102+104+106+...+198+200 =
100+2+100+4+100+6+...+100+98+100+100 =
2+4+6+...+98+100+50*100 =
2550 + 5000 =
7550

[vinviper1]

Thanks guys!!

[VerbalAttack]

Agree with Ekta & Ian...

Can we generalise the approach just in-case if there is a hard question such as, find the sum of all multiples of 7 from 111 to 200, inclusive?

Adopt 2 step approach;
1) Find number of integers (matching criteria)
2) Find the sum

1) If the number of integers is not obvious then use arithmetic progression formula; an = a1 + (n-1)*d

an - last integer (matching critera of multiple of 7) - 196
a1 - first integer (matching critera of multiple of 7) - 112
n - no of integers (which we need to find)
d - factor - 7

196 = 112 + (n - 1) * 7 ==> n = 13

2) To get the sum use this formula; sum = (a1 + an) * (n/2)

sum = (112 + 196) * (13/2) = 2002.

Comments most welcome..

[Ian Stewart]

Comments most welcome..

Yes, your method is perfectly correct. There are several ways to solve these questions, and as long as you have a method you can apply that gets an answer within a minute or a minute and a half, you should be fine. I'll explain how I approach these questions below, but I don't want to suggest that the method below is any better than the method above- it's just different, and doesn't need a formula, but it does require a bit of thought about how to rearrange each sum.

As long as you know how to add

1+2+3+...+49+50

you can add any sum of numbers in an arithmetic progression (i.e. a sequence that increases by the same amount between each term). I'll call this kind of sum an 'evenly spaced' sum, since consecutive numbers are always the same distance apart. The key is to add the smallest and largest (1+50 = 51), and notice that you get the same thing when you add the second largest and the second smallest (2+49=51), and so on.

So
1+2+3+...+49+50 = 51+51+51...+51

And how many 51s are we adding? Well, we need to know how many pairs of numbers there are in the list 1, 2, 3, ..., 49, 50. There are clearly 50 numbers, and therefore 50/2 = 25 pairs. So

1+2+3+...+49+50 = 51+51+51...+51 = 25*51 = 1275.

I'd guess the above is already well known to most GMAT test takers. We can, however, rewrite any evenly spaced sum so that it starts at 1 and goes up by 1 each time- just like the above. To take the example from VerbalAttack's post:

What is
112+119+126+...+189+196 = ?

We can factor out a 7:

= 7*(16+17+18+...+27+28)

Now, rewrite each term in brackets as "15+x" so the sequence can start from 1, not from 16:

=7*[(15+1) + (15+2) + (15+3) + ... + (15+12)+(15+13)]
=7*(15n + 1+2+3+...+12+13)

where n is the number of terms in the original sequence. Fortunately, at this stage it is clear that n is 13. Now, using the same principle as was used above, when adding 1+2+...+49+50:

=7*(15*13 + 14*(13/2))
=7*286
=2002

I prefer not to rely on formulas, for many reasons, so the above method suits me best, and it's very quick once you practice it. But for the GMAT there are many ways to solve these types of questions, and test-takers should choose the method they're most comfortable with.

[VerbalAttack]

great Ian... good way to rely on the fundamentals than relying on the formulas... t

cheers