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4. 小明在计算 1−2+3−4+5−6+…+49−50 时, 不小心把一个运算符号写锴了 (“+"错写成“ - "或“-"锴写成“+")，结果算...
Mar 17, 2024
4. 小明在计算 1−2+3−4+5−6+…+49−50 时, 不小心把一个运算符号写锴了 (“+"错写成“ - "或“-"锴写成“+")，结果算成了 -67 ,则原式从左往右数, 第个运算符号写错了.
Solution by Steps
step 1
Consider the sum of the series without any mistakes: S=1−2+3−4+…+49−50. This is an alternating series with 50 terms
step 2
The sum of the first n positive integers is given by the formula 2n(n+1). Therefore, the sum of the positive integers from 1 to 50 is 250⋅51=1275
step 3
The sum of the first n negative even integers is −2−4−6−…−50=−2(1+2+3+…+25). Using the formula from step 2, this sum is −2⋅225⋅26=−2⋅325=−650
step 4
The correct sum of the series should be the sum of the positive integers minus the sum of the negative even integers: S=1275−650=625
step 5
Since the student's incorrect result was −67, we need to find the difference between the correct sum and the incorrect sum: 625−(−67)=625+67=692
step 6
The difference of 692 must be twice the value of the number at the position where the mistake occurred because changing a "+" to a "-" or vice versa affects the sum by twice the value of the number (once for removing the original sign and once for adding the opposite sign)
step 7
To find the number where the mistake occurred, we solve 2⋅n=692 for n. Dividing both sides by 2 gives n=346
step 8
Since the series alternates between positive and negative numbers, and the mistake must have occurred at an even position for the sum to be negative, we divide 346 by 2 to find the position of the operator that was written incorrectly: 346÷2=173
Answer
The mistake occurred at the 173rd operator.
Key Concept
Identifying the position of a mistake in an alternating series
Explanation
The position of the mistake in the series was found by calculating the difference between the expected sum and the incorrect sum, then determining that the mistake must have occurred at an even position to result in a negative sum.
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