A sequence consists of 16 consecutive even integers written

Question

A sequence consists of 16 consecutive even integers written in increasing order. The sum of the first 8 of these even integers is 424. What is the sum of the last 8 of the even integers?

A. 488
B. 540
C. 552
D. 568
E. 584

Zarrolou · Accepted Answer

Actually I have found a QUICK way, check this out.
the first 8 can be written as: 
 x , x+2,...,x+14 
the second 8 can be written as
 x+16,x+18,...,x+30 
As you notice, there is a gap of 16 between each number and its corrispondent 8 positions after 1st=x 9th=x+16
So if we add the sum of the gaps 16*8 to the sum of the numbers without the gap = 424 we have the result
16*8+424=552

Zarrolou · Answer

the sequence can be expressed as:  2x, 2x+2,2x+4 ,...  and generally if P=position(1,2,...) each term is  2x+2(p-1) 
The sum of the first 8 will be  8*2x+2(1+2+3+4+5+6+7)=8*2x+2*28=424  so  8x=396  and  2x=46  so the first number is  46 .
The sum of the last 8 will be from 9th place to 16th place
so  46+2(9-1)+...+46+2(16-1)= 46*8+2(8+9+10+11+12+13+14+15)=46*8+92*2=552 
C

PS:I used 2x as term because I wanted to be sure it was even
Scroll down for my quick solution

Narenn · Answer

Let the first term be n

sequence for first 8 terms n, n+2, ................n+14

Sum = 424 = 8n + (0 + 2 + 4 + 6 + 8 + 10 + 12 + 14) -----------> 424 = 8n + 56 ---------> 8n = 368 ---------> n=46

So the first term is 46

last 8 terms will be n + 16 .........................n + 30 i.e. 62, 64, ..................,76

This is an Arithmatic Progression with first term = n + 16 last term = n + 30 number of terms = 8

Sum =  \frac{number of terms(first term + last term)}{2}

Sum =  \frac{8(62 + 76)}{2}

Sum =  \frac{8(138)}{2}

Sum = 138 X 4 = 552

Regards,

Narenn

Narenn · Answer

Thats it!

Actually i tried very hard to recall this, but i couldn't, as i learned it long back Zarrolou, thanks for reminding this. +1 by heart :-D

Narenn

nt2010 · Answer

The fastest way is to use the sum property of evenly spaced sets

(\frac{First Term + Last Term}{2}) * No of terms  = Sum of the series

For the 1st eight set - (\frac{2n + 2n + 14}{2})*8  = 424. n = 23

For the 2nd eight set - (\frac{2n + 16 + 2n + 30}{2}) * 8  = 552 - which is the answer - C.

Remember, for this problem you can write out the sequence such as 2n, 2n+2, 2n+4, 2n+6 or you can use some formula - 2n + 2 (m-1).. to get the eight term in the sequence you will be 2n + 2*(8-1) = 2n + 14 and the 16 term in the sequence is 2n + 2*(16-1) = 2n + 30. For me the earlier part is quick and less error prone.. while the formula abstraction might be good for you.

//Kudos please, if this explanation is good

atalpanditgmat · Answer

Thnaks Zarrolou, I am still lost somewhere. Am i missing anything basic mathematics in your explanation. Please simplify further.

Transcendentalist · Answer

Approach i used..

Sum of n even consecutive integers = n(n+1)

Sum of 8 consecutive even integers that start at some point after n = (n+8)(n+9)

Given => (n+8)(n+9) - n(n+1) = 424

16n +72 = 424 - General equation for sum of consecutive 8 digits where n is where counting starts

Solving we get n = 22 (Where the count starts)

Sum of 8 consecutive integers after n = 22 will be count starting at n1 = 22+8 = 30

So Answer = 16*30 +72 = 552

To break this down into a formula...

Sum of x consecutive even integers = 2xn + x(x+1)/2 (n = (First term of series/2) - 1)
 
Also for reference sakes

Sum of x consecutive odd integers = (x+n)^2 - (n)^2 (n=(first term of series - 1)/2)

I derived this so dunno if there is a better form of it.

Zarrolou · Answer

Hi atalpanditgmat, let me explain my method.

We have 16 numbers even and CONSECUTIVE so they can be expressed as:  x, x+2, x+4, x+6, x+8, x+10, x+12, x+14, x+16, x+18, x+20, x+22, x+24, x+26, x+28, x+30 
Here they are. Now look at this:  evrey number in the first 8, has a gap of 16 to its corrispondent 8 positions after . That's the trick.
ie: first is  x  => 9 pos after =>  x+16 
second is  x+2  => 9 pos after => x+2+16 
So every number in the last 8 can be written as  1st+16 ,  2nd+16  and so on.
The sum of 1st 2nd 3rd 4th ... 8th is  424 , so the sum of 9th 10th ... 16 th or (using the trick)  1st+16, 2nd+16, 3rd+16, 4th+16, ... 8th+16  is  424 (the sum of the numbers without 16)+ 16*8 (the sum of 16s)

Let me know if it's clear now

atalpanditgmat · Answer

Aha! I get the right path now. Thanks thanks a lot....

mau5 · Answer

For any given AP : Sum of the n terms : n/2*(A1 + An) and An = A1+(n-1)d, where d = common difference,(2 in this case)
Given : 8/2*(A1+A8) = 424 
Now the sum of the last 8 terms = 8/2*(A9+A16)--> 4*(A1+8d+A1+15d) = 4*  = 424+64*2 = 424+128 = 552

C.

umg · Answer

Another Method

Sum of 1st 8 Terms = 424
Average of those 8 Terms =  \frac{424}{8} = 53

i.e. 4 Even Terms after 53 from the first part of series.
Therefore, 4Th Term = 54
8th Term = 54 + 3 * 2 = 60

9th Term = 62
Sum from 9th till 16th Term = 62+64+66+68+... 
= 60*8 + 2+4+6+8+... +16 = 480 + 2(1+2+3+...+8) = 480 + 2*  \frac{(8*9)}{2}  (Sum of consecutive numbers from 1 to n =  \frac{(n)*(n+1)}{2} )

= 480+72 = 552 (C)

gauraha · Answer

(a1+a8)/2*8=424
a1+a8=106
since nos. are consecutive integers a4+a5=a1+a8
No. between a4 and a5 is (a4+a5)/2=53
So a4=53-1=52
First term is 52=a1+(4-1)*2
a1=46
Eigtth term from last is 9th term from first
So a9=46+(9-1)*2=62
and 16th term a16=46+(16-1)2
 a16=76
 So sum from ninth term to 16th term =(62+76)/2*8 =552

LeonidK · Answer

The answer is C
first 8: 424
last 8: 424 + 8 * 16 = 552

bumpbot · Answer

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A sequence consists of 16 consecutive even integers written

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