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

Question is how we can solve such complex algebraic equation in short time.

I suggest POE method.

1. Choose option C, put this value in the equation (First selection is based on instinct )
=> (49+43)(49-42) = 744
=> 92 * 6 = 744
=> 553 = 744
Here we can conclude that n should be greater than 49 . Options A,B and C are eliminated

2. Chose option D, put this value in the equation
=>(50+43)(50-42) = 744
=> 93 * 8 = 744
=> 744 = 744 Bingo !!! => n is 50

Is there any other method available?

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

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

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

Thanks guyz for the various ways. OA is D
pray the mind in open enough during the actual exam to be able to think out benchmarking and the other techniques!

Here's some more addition from Bunuel..
Look at answer choices:

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

n-42
n+43

It can not be A as 47-42=5 and 744 is not a multiple of 5.
It can not be B as 48-42=6 and 43+51=91 their multiple will have 6 for the last digit

It can not be C as 49-42=7 and 744 is not a multiple of 7.
It can not be E as 51-42=9 and 744 is not a multiple of 9.

n is the number of terms so it must be positive. It must be greater than 42 to ensure that (n - 42) is not negative.

744 needs to be written as a product of two numbers. Factorize 744.
You get 744 = 8*93 = 8*3*31
We need one of the numbers to be greater than 85 to account for (n+43) where n is greater than 42.
We see 8*93 is possible. If the (n+43) factor is 93 but n must be 50.

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

For the number of terms in the set of consecutive integers, it doesn't matter whether the sum is positive or negative.

If we are told that integers from a to b are inclusive then the # of terms would be: last term (biggest)-first term(smallest)+1=b-a+1.

eg. set from -42 to -35 inclusive: -35-(-42)+1=8.
OR
set from -59 to -42 inclusive: -42-(-59)+1=18
OR
set from -1 to 4 inclusive: 4-(-1)+1=6

When a and b are not inclusive, # of terms between a and b: b-a-1

As explained above n must be larger than 42 to compensate for the negative numbers in the set.

That means that 372 is the sum of x number of consecutive integers >43 (i.e. 43 + 44 + ... + (43+x)), i.e. n = 43+x.

Solving for x:
If we divide 372 by 43, we yield a remainder of 28.

x is the number of consecutive integers, whose sum is 28: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
Thus x is 7 and the n is 43+x = 43+7=50.

The answer is D.

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

Bunel: I understand how you are calculating # of terms given the fact that the sum is +ve. What if the sum were -ve? In that case "n" can be smaller or greater than
-42. How do we calculate # of terms in that case.
Thank you in advance.

We can also use some divisibility properties, because we have a sequence of evenly spaced integers.
If the number of terms in the sequence is odd, the sum of the numbers is a multiple of the middle term.
If the number of terms is even, the total sum is a multiple of the sum of the two middle terms.
If we factorize 372 we get 2*2*3*31. By combining the factors we cannot get a divisor of 372 in the range of 40-50, but 3*31 = 93 = 46+47.
So, we have an even number of terms, with 46 and 47 the two middle terms, first term 43, then the last must be 50.

Not shorter than your solution, but it's fun to play with divisibility rules, sometimes just for the sake of practice...

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

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

I forgot about AP.....
Bunnel you are amazing..........
This is how i tried which is ridiculous as it took arnd 4 mins

Step1: Obviously, we need not worry about numbers till 42 as they will negate with the negative numbers when addition is performed: [strike]-42, -41........0........., 41, 42[/strike], 43...n => 43+44+...+n = 372
Step2: the series can be represented by: 43 + 0, 43+1, 43+2.....43+(x-1)# = 372
# --> Series starting with zero
Step3: 43x + [(x-1)x]/2 = 372 --> Solve quadratic equation.
Step4: x^2 - x + 86x = 372
=> x^2 + 85n -744 = 0
=> x^2 + 93x -8x - 744 = 0
=> (x+93)(x-8)=0
x=8.
Step5: But mind you the series ends with x-1. Hence 43 + (8-1) = 50.

-42 - 0 = 0 - 42 thus 372 must equal 43 - X

If you average out to 50 and than kinda thumb it you see that there are 8 #'s over 42 that = 372 42 + 8 = 50

D

Bunuel, is there a quick way to solve the equation:

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

?

I mean the question is very simple. But the equation looks very tedious to me.

hello bunuel, i do understand the process. but i really want to know is there any short cut for 744=(n+43)(n-42) to arrive n=50

thanks in advance

Denote n - 42 = x, where x is a positive integer, and of course, n + 43 = x + 85.
You have to solve x(x + 85) = 744. Look for the factorization of 744:
744 = 2 * 2 * 2 * 3 * 31 = 8 * 93 (8 * 90 = 720, so you should try factors around 8 and 90)

Therefore, x = 8 and n = 50.