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Problem 20 - Entrance Test
Evaluate the sum: Sum from k=1 to 100 of (1/k - 1/(k+1)).
Correct: B
The sum is a telescoping series. Let's write out the first few terms and the last few terms of the sum:
For k=1: (1/1 - 1/2)
For k=2: (1/2 - 1/3)
For k=3: (1/3 - 1/4)
...
For k=99: (1/99 - 1/100)
For k=100: (1/100 - 1/101)
Now, let's sum these terms:
Sum = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/99 - 1/100) + (1/100 - 1/101)
Notice that the middle terms cancel each other out:
-1/2 cancels with +1/2
-1/3 cancels with +1/3
... and so on.
-1/100 cancels with +1/100
The only terms that remain are the very first term and the very last term:
Sum = 1/1 - 1/101
Sum = 1 - 1/101
To combine these, find a common denominator:
Sum = 101/101 - 1/101
Sum = 100/101.
The value of the sum is 100/101.