Consider the expression 2 + 4 + 6 + 8 + ... + 78 + 80. What

Question

Consider the expression

2 + 4 + 6 + 8 + ... + 78 + 80.

What is the smallest number of plus signs that must be changed to minus signs so that the value of the resulting expression is a three-digit negative number? (The sum of the n smallest positive integers is n(n+1)/2.)

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

paragw · Answer

Here's my solution

AdarshSambare · Answer

The key to solution is understand we need to replace the sign, which will lead to multiplication of 2 from original sum of n number.

We need to change plus signs to minus signs, so each changed term changes the total by twice that term.

Original sum:
2 + 4 + ... + 80 = 2(1 + 2 + ... + 40)
= 2 × 820 = 1640

If we change some plus signs to minus signs, we subtract 2 × (sum of the corresponding numbers).

Taking help from options:
I started with 9 because it was easier but understood quickly that the answer is either 11/13.

Try 11 signs
To get the largest decrease, choose the 11 largest terms:
60 + 62 + ... + 80
Their sum is:
11/2 × (60 + 80) = 770
So new value:
1640 − 2(770)
= 1640 − 1540
= 100

Not negative.

Try 13 signs
Choose the 13 largest terms:
56 + 58 + ... + 80
Sum:
13/2 × (56 + 80) = 884
New value:
1640 − 2(884)
= 1640 − 1768
= −128

This is a three-digit negative number.

Therefore, the smallest number of plus signs that must be changed is:

Answer: (E) 13

kevincan · Answer

We could also divide everything by two

chetan2u · Answer

A different way of looking at the question...

Sum =  2+4+...+80 = 2(1+2+...+40) = 2(\frac{40×41}{2})=1640

When will the sum be 0?
 When the positive signed and negative signed both are 820 each.

Now when will the sum be at least -100?
 When we subtract 50 from positive signed and add 50 to negative signed.

We should concentrate on the positive terms as we know that these are continous integers from 2.. 
  So, Sum of positive terms should be less than or equal to 820-50 or 770..

Sum =  2+4+...+2n = 2(1+2+...+n) = 2(\frac{n(n+1)}{2} ≤770
 n(n+1) ≤770
Now 27*28 =756 while 28*29 > 770
So, maximum number of positive signs =27

Thus,  minimum number of minus signs will be 40-27 or 13. (Total terms are 40)

E

Consider the expression 2 + 4 + 6 + 8 + ... + 78 + 80. What

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GMAT Problem Solving (PS) Questions

Consider the expression 2 + 4 + 6 + 8 + ... + 78 + 80. What

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