In the first week of the Year, Nancy saved $1.

The sum of first n consecutive positive integers is given by the formula n(n+1)/2.
For GMAT, it is a good idea to know this formula. It could simplify many calculations. Test prep companies discuss all such useful formulas in their curriculum. The Official Guides only give practice questions.

You should also know the more generic formula of sum of an Arithmetic Progression. From that you can easily derive this formula.

Sum of an Arithmetic Progression will be n*Average where n is the number of terms in the AP and Average will be the average value of the terms. The average value can be found as (First term + Last term)/2

If first term is a, last term is a + (n-1)*d where d is the common difference.

Average = (a + a + (n-1)*d)/2

Sum = n*(a + a + (n-1)*d)/2 = n/2(2a + (n-1)*d)

In case of consecutive integers starting from 1, a = 1 and d = 1

Sum = n/2(2 + (n-1)) = n(n+1)/2

I used:
Sum = (Average of terms) x (# terms)

Average = (1+52) / 2 = 53/2

# terms= 52-1 + 1 = 52

Sum= 53*52/2 = 53*26 = 1378
Answer C.

Video solution from Quant Reasoning:
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Hi,
This can be solved using the formula:

Sum= n/2( a + nd)

n=52


a= first term=1

d= difference =1

Sum=52/2(1+52)
=1378

For an even list of consecutive numbers where a = the first number and z = the last number and (in this case) a total of 52 consecutive numbers-- (a+z)(.5*number of consecutive integers)---> (1+52)*26=1378

Simply using summation of first natural numbers formula = n(n+1)/2

=(52*53)/2 = 1378

Hence OA: C

How did you know about that formula (n(n+1))/2 ? I've read through the GMAT Official Guide Math Review but don't recall learning that. It seems if you know that formula the question is very easy, but if you don't the problem can be a big time drain. Do you recommend other guides I should be reading to study up on tips and formulas for problems like these?

Yes! Definitely better/easier/faster to apply logic than to memorize another formula. This way, you can use the logic even if you don't start at one. Or if you are counting by 2's or 3's. For example, what if Nancy saved $51 the first week of the year, and increased her savings by $1 each week after that.

Start at $51. Know that there are 52 weeks in a year, so the last week she will save 102. So the sequence will be 51, 52, 53... 100, 101, 102.
See how adding the first and last is the same as adding the second and the second-to-last, and the same as the third and third-to-last? All like pairs are $153.
If there are 52 items, then there are 26 pairs.
$153 x 26 = $3978

Counting by 3's:
First week is $51. Last week will be (52 - 1) x 3 = 153 more, or 51+153=204. So: 51, 54, 57... 198, 201, 204.
First + last: 51 + 204 = $255.
Times 26 pairs: $255 x 26 = $6630

Counting by 3's, odd number of weeks:
Let's say Nancy only saved for 51 weeks, because she bought holiday presents for everyone in the last week.
First week is $51. Last week will be (51 - 1) x 3 = 150 more, or 51+150=201. So: 51, 54, 57... 195, 198, 201.
First + last: $51 + $201 = $252.
Now, there are 25 pairs + 1. The one is in the very middle. So $252 x 25 = $6300. Still have to add that last number that wasn't part of a pair. What was it?
Nancy saved for 51 weeks, so that's 25 weeks on one side, 25 on the other, the middle is the 26th week.
26th week is (26 - 1) x $3 = $75 more than week 1, or 51 + 75 = 126.
$6300 + $126 = $6426

It's certainly better to apply logic than just learn up formulas for specific situation because you may find you have no formula for a given situation in the question. Also, you need to know exactly when the formula is applicable for example here you must know that the formula is applicable for first n positive integers only.

That said, n(n+1)/2 is a very basic and useful formula. You should know it for GMAT, not because you may not be able to do a question without it but because you may spend unnecessary amount of time on a question when other people will be able to run away on it with the formula.

This is an evenly spaced set since the difference between terms is always 1.

Calculate the number of terms: 52 (last) - 1 (first) + 1 = 52
Calculate the mean: (52+1) / 2 = 26.5
Multiply: 26.5 * 52 = 1,378

Let's first set up the pattern of Nancy's savings. The first week she saved $1, the second week she saved $2, the third week she saved $3, and so forth. Therefore, the total amount of money she will have saved at the end of 52 weeks will be: $1 + $2 + $3 + $4 + … + $52. The pattern is obvious, but the arithmetic looks daunting because we need to add 52 consecutive integers. To shorten this task, we can use the formula: sum = average x quantity.

We know that Nancy saved money over the course of 52 weeks, so our quantity is 52.

To determine the average, we add together the first amount saved and the last amount saved and then divide by 2. Remember, this technique only works when we have an evenly spaced set.

The first quantity is $1 and the last is $52. Thus, we know:

average = (1 + 52)/2 = 53/2

Now we can determine the sum.

sum = average x quantity

sum = (53/2) x 52

sum = 53 x 26 = 1,378

Answer is C.

Note: If we did not want to actually multiply out 26 x 53, we could have focused on units digits in the answer choices. We know that 26 x 53 will produce a units digit of 8 (because 6 x 3 = 18), and the only answer choice that has a units digit of 8 is answer choice C.

APPROACH #1: No formula required
We want to add 1+2+3+4+...+51+52
So, let's add them in pairs, starting from the outside and working in.
1+2+3+4+...+51+52 = (1+52) + (2+51) + (3+50) + . . .
= 53 + 53 + 53 + ....

How many 53's are there in our new sum?
Well, there are 52 numbers in the sum 1+2+3+..+52, so there must be 26 pairs, which means there are 26 values in our new sum of 53 + 53 + 53 + ....

So, what is (26)(53)?
Fortunately, if we examine the answer choices, we see that we don't even need to calculate (26)(53)

Why not?
Notice that when we multiply (26)(53), the units digit in the product will be 8 (since 6 times 3 equals 18).

Since only 1 answer choice ends in 8, the correct answer must be C

APPROACH #2: Apply formula
The formula: 1 + 2 + 3 + . . . + n = (n)(n+1)/2
We want to add 1 + 2 + 3 + 4 + ... + 51 + 52, which means n = 52

Plug n = 52 into the formula to get: 1 + 2 + 3 + . . . + 52 = (52)(52+1)/2 = 1378

Answer: C

RELATED VIDEO

Bunuel

I often get confused between n(n-1)/2 and n(n+1)/2 for consecutive integers.

When do we use n(n-1)/2 could you please explain a little.
Thanks in advance.

Check this thread: https://gmatclub.com/forum/sum-of-n-pos ... 63301.html

Hope it helps.

Hi All,

This question describes a sequence of numbers: 1, 2, 3,…..52 and asks you to add them all up.

Adding the numbers up in order would take way too much time to be practical. There are actually a couple of ways to quickly add up these terms; here's one way called "bunching":

When adding up a group of numbers, the order of the numbers does not matter. I can group the numbers into consistent sub-groups:

1+52 = 53
2+51 = 53
3+50 = 53
4+49 = 53
etc.

So every group of 2 terms sums to 53. There are 52 total terms, so that means that there are 26 sets of 2 terms.

26(53) = 1378

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Nancy saved 1$,2$,3$....52$.
Total savings of Nancy =(1+2+3+4+....52)$
= 52x53/2 [As we know that 1+2+3+4+....n=n(n+1)/2]
= 26x53
=$1378 (C)

Hope this helps

I always end up getting confused with the rules and formulas.I ended up doing this question with Sn general formula (n/2 ( 2a + (n-1) d) . But got to do this in two minutes time in real test. So, I hope I am correct with the below.


1- For series in AP , Mean = Median right?
And here the numbers are even so 26+ 27/2 will be the average?

N+ 1 will be the number of terms

Sum = number of term * average

2- Also , Average in series could be found out by first team + last team /2

And multiplying Average with number of terms with be Sum of terms?

I hope, these formula (if applicable) applies to all kind of questions?

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