OG 13 - PS 158

[gdshamain]

158. In the first week of the year,Nancy saved $1.In each
of the next 51 weeks,she saved $1 more than she
had saved in the previous week. What was the total
amount that Nancy saved during the 52 weeks?
(A) $1,326
(B) $1,352
(C) $1,378
(D) $2,652
(E) $2,756

[[email protected]]

Hi gdshamain,

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: C

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

[Brent@GMATPrepNow]

In the first week of the year, Nancy saved $1. In each of the next 51 weeks, she saved $1 more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks?

A. $1,326
B. $1,352
C. $1,378
D. $2,652
E. $2,756

Here's a similar approach with a slight TWIST at the end.

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 does (26)(53) equal?
Fortunately, if we examine the answer choices, we see that we don't 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 (C) ends in 8, the correct answer must be C

Cheers,
Brent

[[email protected]]

Hi gdshamain,

There's also a more-formal arithmetic approach for sequences that are made up of evenly-spaced terms:

1) Take the average of the smallest and largest terms: (1+52)/2 = 26.5
2) Count up the total numbers of terms: 52
3) Multiply the two values: (26.5)(52)

You'll end up with the exact same answer: C

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

[confused13]

If I can add my two cents:

https://en.wikipedia.org/wiki/1_%2B_2_%2 ... _4_%2B_⋯

A general formula:
[n*(n+1)]/2

[gdshamain]

Thank you for the response.

[Jeff@TargetTestPrep]

158. In the first week of the year,Nancy saved $1.In each
of the next 51 weeks,she saved $1 more than she
had saved in the previous week. What was the total
amount that Nancy saved during the 52 weeks?
(A) $1,326
(B) $1,352
(C) $1,378
(D) $2,652
(E) $2,756

Solution:

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: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.

[seandcarey]

Another way to do it is

N*(N+1)/2

In this case N is equal to 52.

(52*53)/2 = 1378

If it were 100
100 * 101 / 2 = 5050

It is very useful for expected value questions also. For example, what is the EV of the roll of a single die.

EV = Sum of outcomes / # of outcomes
6*7 = 42/2 = 21. 21/6 = 3.5