Thank you for the response.

gdshamain wrote: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.

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