🧭 Problem

In the expression \(1+3+5+7+\cdots+97+99\), A boy changes some plus signs to minus signs. The new value is negative. What is the least number of plus signs she could have changed?

1️⃣ Evaluate the original total

There are \(50\) odd terms from \(1\) to \(99\). Using the classic identity:

2️⃣ Make it most negative with the fewest flips

To minimize the number of flips, change the largest terms to negative. Keep the first \(n\) odds positive and flip the rest:

The positive part sums to \(n^2\). The flipped part is the remainder, with sum \(2500-n^2\). So the new value is

We want it negative:

3️⃣ Count the flips

With \(n=35\) positives, the number of flipped signs is the remaining odd terms:

If \(n=36\), then \(2n^2-2500=2592-2500=92>0\) (not negative). Hence the least number of flips is \(15\) → option (B).

📝 Quick Check

Same idea for \(1+3+\cdots+199\) (there are \(100\) terms). What is the least number of flips to make it negative?