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Arthmetic subsequent value Problem Solving question

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by sam mwambazi » Thu Dec 06, 2018 2:21 pm

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Does anyone know how to solve this question with an equation that is easy to understand?
A certain drive-in movie theater has a total of 17 rows of parking spaces. There are 20 parking spaces in the first row and 21 parking spaces in the second row. In each subsequent row there are 2 more parking spaces than in the previous row. What is the total number of parking spaces in the movie theater?
A. 412
B. 544
C. 596
D. 632
E. 692
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by ceilidh.erickson » Fri Dec 07, 2018 1:08 pm
If each row had 2 more spaces than the previous, we could imagine this as [x, x + 2, x + 4, x + 6... x + n]. Because the 1st & 2nd rows have different spacing than the other rows, there's no way to do this in one simple step, but we can divide it into 2 steps.

For this problem, the first & second rows are 1 apart, then everything afterwards is 2 apart. First, it helps to establish your pattern:

1st --> 20
2nd --> 21
3rd --> 23
4th --> 25
5th --> 27

For every term after the 2nd term, we've added 21 + 2(the # of terms between 2 and that term). For example, there are 3 terms between the 2nd to 5th term, inclusive. So 21 + 2(3) = 27.

The 17th term will be the 15th term after the 2nd. So 21 + 2(15) = 51. (Or you could just list them all out. It wouldn't really take that long. In fact, it might be faster than trying to derive the pattern!).

Now, we'll find the sum in 2 steps:
1. find the sum from the 2nd row (21) to the 17th row (51)
2. add the value from the 1st row

To add up all of the odd integers from 21 to 51, we use the CONSECUTIVE SUM formula: sum = (average)(# of terms)
The average of a consecutive set = the average of the endpoints --> (biggest + smallest)/2
The # of terms = the difference between the biggest and the smallest, divided by the step. Then add 1 to make it inclusive. So the # of odd terms --> (biggest - smallest)/2 + 1

So for odd integers from 21 to 51:
average = (21 + 51)/2 = 72/2 = 36
# of terms = (51 - 21)/2 + 1 = 30/2 + 1 = 16
sum = (36)(16) = 576

To get the total for all the spaces, we need to add the 20 from the 1st row: 576 + 20 = 596.

The answer is C.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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by Scott@TargetTestPrep » Thu Apr 04, 2019 5:32 pm
sam mwambazi wrote:Does anyone know how to solve this question with an equation that is easy to understand?
A certain drive-in movie theater has a total of 17 rows of parking spaces. There are 20 parking spaces in the first row and 21 parking spaces in the second row. In each subsequent row there are 2 more parking spaces than in the previous row. What is the total number of parking spaces in the movie theater?
A. 412
B. 544
C. 596
D. 632
E. 692
We can see that the 3rd row has 21 + 2(1) = 23 parking spaces, the 4th row has 21 + 2(2) = 25 parking spaces and so on. Therefore, the 17th row has 21 + 2(15) = 51 sparking spaces. Therefore, the total parking spaces of row 2 to row 17 is:

21 + 23 + 25 + ... + 51

Since the terms of the above sum are evenly spaced, we can use the formula sum = average x quantity. The average is (21 + 51)/2 = 72/2 = 36. The quantity is 17 - 2 + 1 = 16. Therefore, the sum is 36 x 16 = 576. We still have to add the number of parking spaces in the first row, so the total number of parking spaces in all 17 rows is 576 + 20 = 596.

Answer: C

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