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If the sum of the consecutive integers from –42 to n

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Bunuel

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23 Jul 2013, 15:42

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jjack0310

Bunuel

jjack0310

Can you please explain how is the number of terms 42 + n +1?

Thanks

# of term from a to b, inclusive is b-a+1.
# of term from -42 to n, inclusive is n-(-42)+1=42+n+1.

OK I get it.

And finally, (N+43)*(N-42) = 744

How is N = 50?

Do you actually solve for N = 50?
Multiply the two terms and subtract 744 or a easier and simpler way?

Thanks

You can plug answer choices starting from C and if it won't be correct, then go up or down.

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30 Sep 2013, 22:45

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It is consecutive integers,

So, -42...-41...-40....-1...0...+1...+2....+42...(+43...+44....+49...+50)
Sum up 43 to 50=372
n=50

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01 Oct 2013, 05:04

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hi all, it becomes quite an easy question if we know the below basic formulae

nth term in A.P = l = a+ (n-1)d where l = last number in series, a= first number in series, n = total number of terms in series and d = diff
sum of n terms = (a+l)*n/2

Now applying the above
a= -42
d =1
Sum = 372
last term = l = -42 + (n-1) = n-43
sum = (-42 +n-43) *n/2 = 372
= (n-85)*n = 744
=\(n^2\)-85n-744 = 0
(n-93)(n+8) =0
n can't be -8 and hence n =93

hence the last term here = l = a + (n-1)d = -42 + (93-1)*1 = -42+92 = 50

ankur1901

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01 Oct 2013, 05:09

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monirjewel

It is consecutive integers,

So, -42...-41...-40....-1...0...+1...+2....+42...(+43...+44....+49...+50)
Sum up 43 to 50=372
n=50

hi,
how do we determine below? or how do we know that we need to stop at 50 to get 372

Sum up 43 to 50=372
numbers can be weird and it may go upto 43 to 93 or any other number...

KarishmaB

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01 Oct 2013, 20:01

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ankur1901

monirjewel

It is consecutive integers,

So, -42...-41...-40....-1...0...+1...+2....+42...(+43...+44....+49...+50)
Sum up 43 to 50=372
n=50

hi,
how do we determine below? or how do we know that we need to stop at 50 to get 372

Sum up 43 to 50=372
numbers can be weird and it may go upto 43 to 93 or any other number...

Give it a little more thought. How can adding consecutive numbers from 43 to 93 give you a total of 372? In this context, 372 is not a big numbers. You just need a few numbers from 43 onwards to make a sum of 372.

43 + 44 + 45 + 46 + 47 + 48 ... adding a few numbers with an approximate average of 45 gives us a sum of 372.
45*8 = 360 which is close to 372.

So 43 onwards we need only 8 numbers: 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50

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02 Oct 2013, 10:02

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# of term from a to b, inclusive is b-a+1.
# of term from -42 to n, inclusive is n-(-42)+1=42+n+1.

thanks Bunue,,, I was looking for the # of term formula, because it is not in the Math Book.

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05 Oct 2013, 18:12

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Bunuel

# of terms =\(42+1+n=(n+43)\)

\(Sum=372=(n+43)*\frac{(n-42)}{2}\)

\(744=(n+43)*(n-42)\)

\(n=50\)

OR

42 terms after zero and 42 terms below zero will total 0. So, our new question will be consecutive integers with first term 43 have sum 372, what is the last term:

\(\frac{43+n}{2}*(n-43+1)=372\)

\((n+43)*(n-42)=744\)

\(n=50\)

Answer: D (50)

Bunuel will you be kind enough to explain this manipulation, I can't seem to get it done!
\((n+43)*(n-42)=744\)

Thanking you in advance.

Bunuel

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06 Oct 2013, 03:55

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suk1234

Bunuel

Bunuel will you be kind enough to explain this manipulation, I can't seem to get it done!
\((n+43)*(n-42)=744\)

Thanking you in advance.

Multiply \(372=(n+43)*\frac{(n-42)}{2}\) by 2 to get \(744=(n+43)*(n-42)\).

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07 Oct 2013, 17:42

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Using AP formula

Sn =\(\frac{n}{2}(2a+(n-1)d)\)
\(372\) = \(\frac{n}{2}(2(-42) + n-1)1)\)
\(n^2-85n-744\)\(=0\)
\((n-93)(n+8)\) \(=0\)
\(n =93 or n = -8\)

What am i missing?

KarishmaB

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07 Oct 2013, 20:45

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GMAT40

Using AP formula

Sn =\(\frac{n}{2}(2a+(n-1)d)\)
\(372\) = \(\frac{n}{2}(2(-42) + n-1)1)\)
\(n^2-85n-744\)\(=0\)
\((n-93)(n+8)\) \(=0\)
\(n =93 or n = -8\)

What am i missing?

Everything till here is correct. n cannot be negative so n must be 93. So basically we are adding 93 terms starting from -42 to get the sum of 372.

-42 -41 - 40 .... 93 terms
First term - 42
Second term - 41 (you obtain by adding 1 to -42)
Third term -40 (you obtain by adding 2 to -42)
93rd term 50 (you obtain by adding 92 to -42)
So last term is 50.

OR

Last term = a + (n-1)*d = -42 + (93 - 1)*1 = 50

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04 Oct 2015, 08:48

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Bunuel

gmattokyo

Nice solution. Correct as always.
I was actually thinking of having a ballpark and reach the answer from 43 to 50.
Add from 43, 44, 45.. to reach 372, need at least 7 (43x7=301) and less than 9 (43x9=387).

Did you use any intermediary formulas for this part?
(n+43)*(n-42)=744
n=50

Sum of consecutive integers or sum of terms in any evenly spaced set (AP): \(\frac{first term+last term}{2}*number of terms\)

In original question:
First term: -42
Last term: n
Number of terms: 42+n+1=n+43

\(\frac{(n-42)}{2}*(n+43)=372\)

\((n-42)*(n+43)=744\)

please clarify the method the sum of consecutive integer when the first term is not 1

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04 Oct 2015, 08:56

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Bunuel

gmattokyo

If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

i did not find this rule in you math book, please clarify the rule

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04 Oct 2015, 09:24

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anik19890

Bunuel

gmattokyo

If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

i did not find this rule in you math book, please clarify the rule

Post your query once. You posted 2 related posts in less than 30 minutes.

As for your question, sum of n terms of an arithmetic progression (or a sequence in which the difference between consecutive terms is equal) is given by:

S= (n/2)*[2a+(n-1)*d]

where n= number of terms, a=first term, d=difference between consecutive terms

Alternately, the same rule becomes , S= (first term+last term)/2 or in other words, the sum is equal to the average of the first and the last term of an arithmetic progression.

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20 Oct 2015, 11:11

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gmattokyo

If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

My Solution:

Digits from -42 to + 42 will be cancelled out, therefore there sum becomes 0 and first digit would me 43.

Minimum value of "n" as per answer choices is 47 and maximum value of "n" is 51.

So let's try digit 51,

43+44+45+46+47+48+49+50+51=423, but required sum is 372.

Let's try 50, 43+44+45+46+47+48+49+50=372, Answer is Option D

For addition we can also use formula [b](Last term + first term)/2 * Number of terms[/b]

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20 Oct 2015, 11:16

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gmattokyo

If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

My Solution:

Digits from -42 to + 42 will be cancelled out, therefore their sum becomes 0 and first digit would me 43.

Minimum value of "n" as per answer choices is 47 and maximum value of "n" is 51.

So let's try digit 51,

43+44+45+46+47+48+49+50+51=423, but required sum is 372.

Let's try 50, 43+44+45+46+47+48+49+50=372, Answer is Option D

For addition we can also use formula [b](First term + Last term)/2 * Number of terms[/b][/b]

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11 Mar 2016, 19:52

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gmattokyo

If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

2 methods...
we know for sure that n must be > 42.
we can test numbers.
so -42 up to 42, will equal to 0.
now we have:
43, 44, 45, 46, 47, 48, 49, 50, 51 - let's test...
43+47 = 90
44+46 = 90
45+48 = 93
total: 273..we are short 99..which must be 49 and 50..so n must be 50.

second approach...
find sum of the first 42 numbers
we can do so by applying the formula: n(n+1)/2.
42*43/2 = 21*43 = 903. but since we have -42, it must be true that this is -903.
now..n is greater than 42...and when the sum of all positive and negative is 372, then the sum of only positive numbers would be: 903+372 = 1275.
n(n+1)=1275
since we need a number that either ends in 5 or if divided by 2 has the last digit 5, we an eliminate
A (has 7 as units digit, and 48 - 8 as units digit)
B (has 8 as last digit, and 49 - 9 as last digit)
C (has 9 as last digit, and 50/2 - 5 as last digit) - hold
D if divided by 2 has 5 as the last digit - hold
E no.

now..between C and D...it doesn't take long to see that C is not working...
49x50/2 = 49x25 = (50-1)x25 = 1250 -25 = 1225. but we need 1275. so out.

D - 50x51/2 = 25*51 = 1275.

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17 Mar 2017, 11:31

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If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?

A. 47
B. 48
C. 49
D. 50
E. 51

Another approach, process elimination:

Let's try D 50

Sum = 50 - -42 = 92 + 1 = 93

S = n/2(a+l)
93/2(50 - 42)
41.5(8)
= 372

Therefore, the answer is D

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25 Mar 2017, 20:31

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My approach:
-42 + -41 + .... + 42 = 0
43 + 44 = 87 + 45 = 132 + 46 = 178 + 47 = 225 + 48 = 273
273 + 49 = 322 + 50 = 372

Therefore, the answer is D

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28 Mar 2017, 12:12

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Pretty interesting question
Here's how i solved it
sum of -42 to 42 is equal to 0
Now start from 43
Start plugging in the answer choices
Start with c

43 to 49
Sum from 43 to 49
Number of integers from 43 to 49
=49-43=6+1=7
Sum=7*(49+43/2)
=7*46=322

Since 322 is lesser than 372 hence options a,b and c can be ruled out.
Since we know that sum of integers from 43 to 49 is 322
If we add 50 to it , we get our answer
322+50=372

Therefore the correct option is "D"

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01 Apr 2017, 17:45

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Neglect numbers from -42 to +42, because the summation of this series is positive

Start from 43

\(Sum = [2*43 + (n-1)] * \frac{n}{2}\)
\(744 = 86n + n^2 - n = n^2 +85n\)

\(744 = n^2 + 85n\)

Now substitute : {answer choice - 43 + 1} in above equation

50 it is

OR

\(Sum = (first term + last term) * \frac{no of terms}{2}\)

RHS = (43 + answer choice) * (answer choice - 43 + 1)/2