Find the the sum of the first 20 terms of this series which

Question

Find the the sum of the first 20 terms of this series which begins this way : -1^2 + 2^2 - 3^2 + 4^2 - 5^2 + 6^2...

(a) 210 
(b) 330 
(c) 519 
(d) 720 
(e) 190

Show working please!

anindyat · Accepted Answer

Getting a

this is pair of a^2 - b^2 
after breaking it in (a+b)(a-b)

Sum = (1+2) + (3+4) +...(19+20)
Sum = 210

MBAlad · Answer

I'm missing something here -1^2 = 1, 2^2 = 4 -3^2 = 9

So I'm getting 1 + 4 + 9...........?????

anindyat · Answer

â€“1^2 + 2^2 â€“ 3^2 + 4^2 â€“ 5^2 + 6^2............-19^2 + 20^2
= (20^2 -19^2) + (18^2 -17^2)+...(2^2 -1^2)
= (20 + 19)(20-19) + (18+17)(18-17)+......+ (2+1)(2-1)
= 20 + 19 + 18 + 17.....+ 3

MBAlad · Answer

I'm half way there:

How does (20+19)(20-19) + (18+17)(18-17)+...... turn into 20+19+18.....??

How did you arrive at 210

Urggghhh I'm having a bad day!

Pzazz · Answer

take the first and last digit,
 on applying (a^2-b^2) formula we get
 21(19)-21(17)+21(15)-21(13)+...............-21(1)
 21(19-17+15-13+11-9+7-5+3-1)
 21(2+2+2+2+2)
 21(10)=210.

MBAlad · Answer

Hi pzazz

 

I dont see how you've applied the (a^2-b^2) formula?

21(19) = 399?

Also why did you use 21?????

I will get there!!!!

Hermione · Answer

let me try to explain...

work 2 digits by 2 digits

-1^2 +2^2 = 2^2 -1^2 = (2+1)(2-1)

next...

â€“ 3^2 + 4^2 = 4^2 â€“ 3^2 = (4-3)(4+3)

Hence you get (1+2) + (3+4) ....

kryzak · Answer

Sorry for reviving an old thread, but isn't there an easier way to do this since it's related to arithmetic series? 

We know that 2^2-1^2 = 3
4^2 - 3^2 = 7
6^2 - 5^2 = 11
and so forth. 

So you have a series of 10 numbers, starting with a1 = 3, d = 4, and n = 10.

Sum of this series is  /2 = 210.

FN · Answer

OK read it again..and the symbol is just a (-)

so we have -1^2 +2^2 - 3^2+4^2....

can be simplifyed..to

(a-b)(a+b) here b =1 and a=2;

(2-1)(2+1)+(4-3)(4+3)...basically it becomes 1+2+3....+20

so the sum of the first 20 integers is (20+1)/2 * (20-1)+1=210

A it is.

spacholi · Answer

Thanks for reposting the question. I couldnt make sense of the symbol either. I am new to GMATClub and I must say that this group is more than just impressive. I appreciate the camaraderie the members show to help each other get their best, and hope to contribute whenever I can. In that spirit....

I got the (2-1)(2+1) + (4-3)(4+3) + (6-5)(6+5) + (8-7)(8+7) + ..... + (20-19)(20+19).... part
But then this translates to 3 + 7 + 11 + 15 +...... + 39, which I solved as 10 terms with next term incremented by 4. 
So their sum would be the middle term (average of 19 and 23 = 21) multiplied by 10 = 210.

Using "Sum of n terms of an equally spaced series = middle terms * n" (if n = even, middle term is average of the two middle terms).

shahany · Answer

I believe this has already been said in other ways, but I got lost in some of the terminology and symbols, so I had to to break it down a little more for myself. Hopefully this helps anyone still confused by this.

This is a arithmetic sequence, but only if we look at it in pairs. So (-1^2 + 2^2) is the first pair. (-3^2 + 4^2) is the second pair and so on. The sum in each pair adds four to the previous pair each time as follows:

(-1^2 + 2^2) = 3 (note: 3 is our starting number)
(-3^2 + 4^2) = 7
(-5^2 + 6^2) = 11
(-7^2 + 8^2) = 15
.
.
.
(-19^2 + 20^2) = 39

As soon as we recognize the sequence, we can use the sum of n terms arithmetic progression formula:
Sum of n terms = (n/2) x (value of 1st term + value of last term)
substitute:
sum of n terms = (10/2) x (3+39) = 5 x 42 = 210

Note: we used 10 as the N # of terms because we turned the 20 original terms into 10 pairs.

For those concerned about speed, on this problem all that we have to do is calculate the first couple of pairs until we see the pattern, then calculate the last pair (-19^2 + 20^2), add it to the outcome of the first pair (-1^2 + 2^2) and multiply by 5. This can be done in way under 2 minutes.

Hope that helps.

kobra92 · Answer

if you didn't see that grouping you could always after doing a couple of numbers see that after every addition step which happens to coincide with even numbers there is an overall increase but the value never attains the value of the squared term added...

so you know that after adding 20^2 the answer will be less than 400

at this point, either guess or do a few terms and see that the negative value build up to higher than 70.....ie 400-330.....

EMPOWERgmatRichC · Answer

Hi sagarag,

This question has a great "visual component" to it, so I'm going to give you some hints and let you try this question again...

First, let's deal with -1^2 + 2^2

1) 2^2 is the equivalent of a 2x2 square. Draw it and include the 4 individual boxes.
2) -1^2 = -1; Draw a line through one of the 4 squares you just drew. You now have 3 squares left. Notice the pattern in the drawing....

3) Try these same steps again with -3^2 + 4^2; you should end up with a larger drawing but the SAME pattern. How many squares are left here?

4) Can you figure out how many squares would be left with -5^2 + 6^2 WITHOUT drawing the picture this time....? And what about the other 'pairs' up values up to -19^2 + 20^2?

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

mvictor · Answer

i got to A..but my main concern,   does PEMDAS apply here at all or not?  
if not, it needs to be specified so.
PEMDAS - parenthesis, exponents, multiplication, division, addition, subtraction. 
exponents come first, so any NEGATIVE integer squared is a positive number...no?
my approach, though lengthy:
list all squares (good to remember squares of the first 20 integers):
1 is positive
4 positive
9 negative
16 positive
25negative
36positive
49negative
64positive
81negative
100positive
121negative
144positive
169negative
196positive
225negative
256positive
289negative
324positive
361negative
400positive

now 
400-361 = 39
324-289= 35
256-225=31
196-169=27
144-121=23
100-81=19
64-49=15
36-25=9
16-9=7
and +14=5
now 39+35+31+27+23+19+15+9+7+5
group to be easier:
35+15=50
27+23=50
31+39=70
19+9=28
7+5=12

last digit is 0, so C is out
50+50+70+40=210
A

p.s. surprisingly or not, the sum of all numbers between 1 and 20, inclusive is 20(21)/2 = 210.

chetan2u · Answer

Hi,
 
 Yes exponents come first...
But -3^2 is not SQUARE of -IVE integer, it is -IVE of square of positive integer..
so you will first square it and then add -ive sign to it.....  -3^2= -9 
If it were (-3)^2, this is what would have been the case given by you..  (-3)^2 = 9

If you simplify  -1^2 + 2^2 - 3^2 + 4^2 - 5^2 + 6^2... +20^2 , you will get 1+2+3+4..+20

stonecold · Answer

Magnificent Question 
Here using the a^2-b^2 property =>
it literally reduces to 1+2+3+4+5+6+.....20=> Sum =210

gracie · Answer

if t1+t2=3,
t3+t4=7,
and t5+t6=11,
then we can assume a difference of 4
between consecutive term pairs
thus, t9+t10=19
19+4/2=21=median of series
10 term pairs*21 median=210
A

exc4libur · Answer

special identities:  (x^2-y^2)=(x-y)(x+y) 
rewrite the sequence:  (2^2-1^2)+(4^2-3^2)… (2-1)(2+1)+(4-3)(4+3)… (2+1)+(4+3)… 
sum of the sequence:  (2+1)+(4+3)… 1+2+3+4+5…20 
sum of first natural numbers (positive integers):  (n)(n+1)/2 
sum of first 20 positive integers:  (n)(n+1)/2… 20(19)/2=210

Answer (A).

vitif · Answer

20^2-19^2= 39 
18^2-17^2=35 
For each new couple of number you can see that the result is decreasing by 4. 
So: 16^2-15^2=31 
 14^2-13^2= 27 
12^2-11^2=23 and so on. 
Add everything -> 3+7+11+15+19+23...+39= 210

Hope it helps

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