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Sum of Sequence

Hi, I can't figure out how to arrive at the correct answer. Please help.

In the sequence a sub 1 (first term), a sub 2 (second term), a sub 3 (third term)….., a sub 100 (100th term), the k th term is defined by a sub k = 1/k - 1/(k+1) for all integers k from 1 through 100. What is the sum of the 100 terms of this sequence?

The answer is 100/101

Author
Woldekidan Tegegne

Posted May 19, 2014

1 Explanation

3

Gravatar Lucas Fink, Magoosh Tutor

This one is really, really tricky! The best thing to start with is a bit of exploration. Figure our the first few terms of the sequence.

When k=1, then a1 = 1/1 - 1/2 = 1/2
When k=2, then a2 = 1/2 - 1/3 = 1/6
When k=3, then a3 = 1/3 - 1/4 = 1/12

Now, here's the trick: it's very tempting to look at the end result fractions, but there's not a very nice pattern there. And that's what we're looking for—a pattern.

There is a nice pattern in the part that includes the subtraction. Remember that we're looking for the sum of the numbers: 1/2 + 1/6 + 1/12 is ugly.

But (1/1 - 1/2) + (1/2 - 1/3) + (1/3 -1/4) isn't so ugly...

(1/1 - 1/2) + (1/2 - 1/3) + (1/3 -1/4)
1/1 - 1/2 + 1/2 - 1/3 + 1/3 -1/4
1/1 - 1/2 + 1/2 - 1/4
1/1 - 1/4

That same process would work for the big sum:

(1/1 - 1/2) + (1/2 - 1/3) + ... + (1/99 - 1/100) + (1/100 - 1/101)

Everything between 1/1 and 1/101 cancels. We're left with just the ends:

1/1 - 1/101
101/101 - 1/101
(101 - 1)/101
100/101

And we're done!

May 22, 2014 • Comment

Seema Naik

Could you calculate the average of the first and last term, multiplied by the number of terms (100)? I tried that but get the wrong answer.

Oct 23, 2014 • Reply

Lucas Fink, Magoosh Tutor

The problem with that approach, Seema, is that it requires the numbers in the list to be evenly distributed. For example, it would work with (1, 2, 3, 4,...) or (1/4, 1/2, 3/4, 1, ....) or with (20, 60, 100, 140, 180,...), but it doesn't work with numbers that aren't equally spaced. In our list, we have numbers like this: (1/2, 1/6, 1/12) as I explained above. Since the spaces between them are not equal, we can't use the sum of consecutive sequences formula.

Oct 29, 2014 • Reply

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