If the sum of the even integers between 1 and n is 79 x 80

Question

If the sum of the  even  integers between 1 and  n  is 79 x 80, where n is an odd integer, then  n  =

A) 39
B) 79
C) 81
D) 159
E) 161

Bunuel · Accepted Answer

The \ number \ of \ multiples \ of \ x \ in \ the \ range = = \frac{Last \ multiple \ of \ x \ in \ the \ range \ - \ First \ multiple \ of \ x \ in \ the \ range}{x}+1 (Check this: https://gmatclub.com/forum/totally-basic-94862.html and this: https://gmatclub.com/forum/math-number- ... 88376.html) Now, as n is odd, then the last even number between 1 and n will be n-1 and the first even number will be 2. So, there are \frac{(n-1)-2}{2}+1=\frac{n-1}{2} even numbers (multiples of 2) between 1 and n . Next, the sum of the elements in any evenly spaced set is given by: Sum=\frac{first+last}{2}*number \ of \ terms , the mean multiplied by the number of terms; \frac{2+(n-1)}{2}*\frac{n-1}{2}=79*80 ; (n-1)(n+1)=158*160 ; n=159 Answer D P.S. We have this question with different answer choices here: https://gmatclub.com/forum/the-sum-of-t ... 68732.html

KarishmaB · Answer

Points to remember: 
Sum of all integers from 1 to x =  \frac{x(x+1)}{2}

Sum of x odd consecutive integers starting from 1 =  x^2

Sum of x even consecutive integers starting from 2 =  x(x+1)

If the sum of even integers starting from 1 is 79*80, it means it is the sum of 79 consecutive even integers:
2 + 4 + 6 + 8 + ...  (total 79 integers, not integers up to 79)
The 1st term is 2, the second term is 4, ... the 79th term will be 158. 
But since n is odd, it must be 159.

Hence, sum of all even integers from 1 to 159 will be 79 * 80.

Answer (D)

gmatophobia · Answer

The sum of the first  x  even integers is given by  x(x+1)

Note:  x ⇒ Number of even integers

Given:  x(x+1) = 79 * 80

Therefore,  x = 79

The first positive even integer in this series=  2

Assume that the last even integer in this series=  y

The number of terms,  x , is given by -

\frac{y - 2 }{ 2} + 1 = 79

y = 79*2 = 158

Hence,  158  is the last even integer in the series. As  n  is an odd integer,  n = 158 + 1 = 159

Option D

CapnSal · Answer

Use the options. We know the sum is given as 79*80.

Let's try option (E) where n = 160. As per the statement, the sum of all even integers from 1 to 160 is 79*80. So what are the even integers from 1 to 160? 2, 4, 6, 8.....158, 160.

Use Sequences formula to find the number of terms or n

160 = 2 + (n-1)*2. You will get n = 80.

Through the concept of sum of a consecutive sequences, we get {(2+160)*80}/2.

This will be equal to (162*80)/2 = 81*80. Not what is given in our question stem but we are close. Let's try the next closest option.

Try option (D) where n = 159. As per the statement, the sum of all even integers from 1 to 159 is 79*80. So what are the even integers from 1 to 159? 2, 4, 6, 8.....158.

Use Sequences formula to find the number of terms or n

158 = 2 + (n-1)*2. You will get n = 79.

Through the concept of sum of a consecutive sequences, we get {(2+158)*79}/2.

This will be equal to (160*79)/2 = 79*80. Same as the question stem.

Answer = (D).

Why did we start from (E)? The Sum is 79*80, and option (E) has 160 as the n; since it is a consecutive even integer series we tried to see if we can somehow get 80 (160/2=80). Did not work but we were close, so try the next closest one.

DanTheGMATMan · Answer

Straightforward sum of evenly spaced set:

https://www.youtube.com/watch?v=XRL9x8BTWog

AlieuK117 · Answer

(Written on cellphone)
Hi! We solved this problem using both the summation/sequencing equation and a bit of background on number properties:

Given the nature of the question we know we will be dealing with number pairs (even and odd)

- With the sum of the  even  integers being 79 * 80, we should be looking for an answer that holds a 79 as a number pair!

We can work backwards from the answer choices to try and find our 79.

(Number Range) 
Sequencing Equation: last = first + (n-1)d
d=2

A: 1 - 39
39 = 1 + (n-1)2
n = 20
Odd: 20
Even: 19

B: 1 - 79
n = 40
Odd: 40
Even: 39

C: 1 - 81
n = 41
Odd: 41
Even: 40

D: 1 - 159
n = 80
Odd: 80
Even: 79

E: 1 - 161
n = 81
Odd: 81
Even: 80

holds our desired number pairs

If the sum of the even integers between 1 and n is 79 x 80

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If the sum of the even integers between 1 and n is 79 x 80

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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745 with Only Self-Prep and GMAT Club

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Perfect 805 Score Debrief by Julia

My Rewards