2. 外语
  3. The sequence of numbers a1,a2,a3,...,an,... is defined byfor each integer n1. What is the sum of the first 20 terms of this sequ

The sequence of numbers a1,a2,a3,...,an,... is defined byfor each integer n1. What is the sum of the first 20 terms of this sequ

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问题 The sequence of numbers a1,a2,a3,...,an,... is defined byfor each integer n1. What is the sum of the first 20 terms of this sequence?
选项 A、 
B、 
C、 
D、 
E、 
答案B
解析 This question asks for the sum of the first 20 terms of the sequence. Obviously, it would be very time-consuming to write out the first 20 terms of the sequence and add them together, so it is reasonable to try to find a more efficient way to calculate the sum. Questions involving sequences can often be answered by looking for a pattern. Scanning the answer choices and noting that they contain fractions with denominators 2, 20, 21, and 22, and nothing in between, seems to confirm that looking for a pattern is a good approach to try.
    To look for a pattern, begin by adding the first two terms of the sequence.

Now, if you add the first three terms of the sequence, you get

Note that you can simplify the sum by canceling the fraction 1/3; that is, the sum of positive 1/3 and negative 1/3 is 0.

If you add the first four terms, you get

Again, you can simplify the sum by canceling. This time, you can cancel the fractions 1/3 and 1/4.

If you write out the next two sums and simplify them, you will see that they are

Working with the sums makes it clear that this pattern continues to hold as you add more and more terms of the sequence together and that a formula for the sum of the first k terms of the sequence is

Therefore, the sum of the first 20 terms of the sequence is equal to

The correct answer is Choice B.
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