Carcass wrote:
In the sequence \(a_1\), \(a_2\), \(a_3\), • • • ,\(a_{100}\), the \(k_{th}\) term is defined by \(a_k = \frac{1}{k} - \frac{1}{k+1}\) for all integers \(k\) from 1 through 100. What is the sum of the 100 terms of this sequence?
A) \(\frac{1}{10,100}\)
B) \(\frac{1}{101}\)
C) \(\frac{1}{100}\)
D) \(\frac{100}{101}\)
E) \(1\)
Alternate approach:
Calculate smaller sums and LOOK FOR A PATTERN.
First term = 1 - 1/2 = 1/2
Second term = 1/2 - 1/3 = 1/6
Third term = 1/3 - 1/4 = 1/12
If the sequence includes only 1 term, the sum = 1/2
If the sequence includes only the first 2 terms, the sum = 1/2 + 1/6 = 2/3
If the sequence includes only the first 3 terms, the sum = 2/3 + 1/12 = 3/4
Notice the pattern:
The numerator of the sum = the number of terms
The denominator of the sum = 1 more than the numerator
Thus:
Sum of 100 terms = 100/101
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