1 + 2 + + k = k(k+1)/2. m and n are positive integers satisfying n

Question

1 + 2 + … + k = \frac{k(k+1)}{2}.   m  and  n  are positive integers satisfying  n < m . What is the value of  n + (n+1) + … + (m-1) + m ?

A.  \frac{(m+n)(m-n-1)}{2}

B.  \frac{(m+n)(m-n+1)}{2}

C.  \frac{(m-n)(m+n-1)}{2}

D.  \frac{(m-n)(m+n+1)}{2}

E.  \frac{(m+n)(m-n-1)}{4}

RohitGoyal1123 · Accepted Answer

Straight forward AP series question. Since n<m and we are required to find  n + (n+1) + … + (m-1) + m , we can add the sum of  1 +2+...(n-1)  and subtract the same from equation leaving the following

sum of  1+2...+m  - sum of  1+2...+(n-1)

so  \frac{(m)(m+1)}{2}  -  \frac{(n-1)(n)}{2}

Solve and Get Option B.

MathRevolution · Answer

=>

n + (n+1) + … + (m-1) + m

= 1 + 2 + … + (n-1) + n + (n+1) + … + (m-1) + m – (1 + 2 + … + (n-1))

= \frac{m(m+1)}{2} - \frac{(n-1)(n-1+1)}{2} 
 
 = \frac{m(m+1)}{2} – \frac{n(n-1)}{2}

= (\frac{1}{2})(m(m+1) – n(n-1))

= (\frac{1}{2})(m^2 + m – n^2 + n)

= (\frac{1}{2})(m^2 – n^2 + m + n )

= (\frac{1}{2})( (m+n)(m-n) + (m+n) )

= (\frac{1}{2})(m+n)(m-n+1)

Therefore, the answer is B.
Answer: B

adewale223 · Answer

It is an arithmetic progression because the values are increasing by adding one.

Let x be number of terms.

x= m-n+1

Sum of AP = x/2(2a + (x-1)d)

x/2(a +a +(x-1)d)

x/2(a+l)

Therefore

m-n+1(n+m)/2

Answer is B

Posted from my mobile device

AnuK2222 · Answer

((First term + last term)/2 )* no. of terms= sum of consecutive terms
((m+n)/2)*(m+n-1)
answer B

hope this is the correct way?

PSKhore · Answer

Just substitute n = 1 and m = 2 in the equations, the sum should be 3
Only  Option B  satisfies

Alexabisaad · Answer

I tried to solve this using numbers.. Let's say 1+2+3+4 is the sequence and it gives us 4*(4+1)/2 based on the initial equation assuming 4=k
since n<m, then we will start with n=1 and end with m=4... 
I then tried to see which solution had what I wanted since (m+n) = 5 then I need a 4 in the numerator which would be m-n+1 = 4-1+1 = 4
Options C and D are out since m-n would be 3 and from the initial equation we don't have any 3 only 4*5 so these are the numbers I was looking for

Let me know if this is a bad way of solving this could not think of anything else

egmat · Answer

Alexabisaad   your approach is excellent and you got the right answer!  Number-picking is not a "bad way" at all - it's actually a smart GMAT strategy that you executed perfectly.

Strategic Insight - When to Use Number-Picking:

Your instinct was perfect here! Use number-picking when:
 
- The problem has abstract variables throughout all answer choices 
- You can quickly test with small, simple numbers 
- The answer choices have distinct numerical patterns

In this case, testing  n = 1, m = 4  immediately revealed which expression equals  10 . This is  faster  than algebra for many students and equally valid.

You can practice similar summation and sequence problems  here  (you'll find a lot of OG questions) - select  Arithmetic  under  Problem Solving  and choose  Medium  level questions to reinforce this skill.

1 + 2 + + k = k(k+1)/2. m and n are positive integers satisfying n <

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