Question: In the sequence a1, a2, a3, ..., a100 the kth term is defined by ak = 1/k − 1/(k+1) for all integers k from 1 through 100. What is the sum of the 100 terms of this sequence?

A) 1/10,100
B) 1/101
C) 1/100
D) 100/101
E) 1

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

Given is that \(a_k=1/k - 1/k+1\).

a1= 1/1-1/2
a2= 1/2-1/3
a3= 1/3-1/4 \s.... \s.. \s. \sa99=1/99-1/100
a100= 1/100-1/101
All the terms from 1/2 to 1/100 will cancel each other out when we add them all together because they have opposite signs. There will only be the phrases 1/1 and -1/101 left.
So sum = 1/1-1/101= 101-1/101= 100/101

Approach Solution 2:

In order to start, below are the first four terms:
a1 + a2 + a3 + a4 = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5)
All terms cancel out, with the exception of the first and last terms, as can be seen. These two terms are combined to give us 1 - 1/5 = 4/5.
Using this pattern for all 100 numbers results in:
1 - 1/101 = 101/101 - 1/101 = 100/101

D is the correct choice.

Approach Solution 3:

ak=1/k − 1/k+1
a1=1/1 − 1/ ( 1+1)
a2=½ − 1 / (2+1)
a100=1/100 − 1 / (100+1)
Sum = (1/1) - (1/2) + (1/2) - (1/3) ...... (1/100+1/101 )
Sum = 1- (1/101)
Sum = 100/101

D is the correct answer.

“In the sequence a1, a2, a3, ..., a100 the kth term is defined by ak" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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