AMC 10 Daily Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 12 Medium

A sequence of numbers t_1, t_2, t_3, \cdots has its terms defined by t_n=\frac{1}{n}−\frac{1}{n+2} for every integer n \geqslant 1. For example, t_4=\frac 14−\frac 16. What is the largest positive integer k for which the sum of the first k terms (that is, t_1+t_2+⋯+t_{k−1}+t_k) is less than 1.499?

  • A.

    2000

  • B.

    1999

  • C.

    2002

  • D.

    2001

  • E.

    1998

Answer:E

Note that

\begin{aligned} t_1+t_2+t_3+\ldots+t_{k-1}+t_k & =\left(\frac{1}{1}-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{5}\right)+\cdots \\ & +\left(\frac{1}{k-1}-\frac{1}{k+1}\right)+\left(\frac{1}{k}-\frac{1}{k+2}\right) \\ & =\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{k-1}+\frac{1}{k}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\cdots-\frac{1}{k+1}-\frac{1}{k+2} \\ & =\frac{1}{1}+\frac{1}{2}+\frac{1}{3}-\frac{1}{3}+\cdots+\frac{1}{k-1}-\frac{1}{k-1}+\frac{1}{k}-\frac{1}{k}-\frac{1}{k+1}-\frac{1}{k+2} \\ & =\frac{1}{1}+\frac{1}{2}-\frac{1}{k+1}-\frac{1}{k+2} \\ & =1.500-\frac{1}{k+1}-\frac{1}{k+2} \end{aligned}

This means that the sum of the first k terms is less than 1.499 exactly when \frac{1}{k+1}+\frac{1}{k+2} is greater than 0.001.

As k increases from 4 , each of \frac{1}{k+1} and \frac{1}{k+2} decreases, which means that their sum decreases as well. When k=1998, \frac{1}{k+1}+\frac{1}{k+2}=\frac{1}{1999}+\frac{1}{2000}>\frac{1}{2000}+\frac{1}{2000}=\frac{1}{1000}=0.001.

When k=1999, \frac{1}{k+1}+\frac{1}{k+2}=\frac{1}{2000}+\frac{1}{2001}<\frac{1}{2000}+\frac{1}{2000}=\frac{1}{1000}=0.001.

This means that \frac{1}{k+1}+\frac{1}{k+2} is greater than 0.001 exactly when k \leq 1998 and is less than 0.001 when k \geq 1999.

In other words, the sum of the first k terms is less than 1.499 for k=1,2,3,4 as well as for 5 \leq k \leq 1998, which is the same as saying that this is true for 1 \leq k \leq 1998.

Therefore, k=1998 is the largest positive integer for which the sum of the first k terms is less than 1.499.

Related Problems
Problem 10 AMC 10 Daily Practice Round 3
Problem 11 AMC 10 Daily Practice Round 3
Problem 13 AMC 10 Daily Practice Round 3
Problem 14 AMC 10 Daily Practice Round 3
Think Academy Powered by ChatGPT