AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series

Complete problem set with solutions and individual problem pages

Problem 8 Easy

What is the value of 1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots+\frac{1}{1+2+3+4+\cdots+100}?

  • A.

    \frac{200}{101}

  • B.

    \frac{1}{200}

  • C.

    \frac{101}{200}

  • D.

    \frac{1}{101}

  • E.

    2

Answer:A

1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots+\frac{1}{1+2+3+4+\cdots+100} \\=1+\frac{1}{(1+2) \times 2 \div 2} + \frac{1}{(1+3) \times 3 \div 2} + \cdots +\frac{1}{(1+100) \times 100 \div 2} \\=1+\frac{2}{2 \times 3}+\frac{2}{3 \times 4}+\cdots+\frac{2}{100 \times 101} \\= 1+2 \times\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{100}-\frac{1}{101}\right) \\= 1+2 \times\left(\frac{1}{2}-\frac{1}{101}\right) \\= 1+ \frac{99}{101} \\=\frac{200}{101}

Related Problems
Problem 4 AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series
Problem 5 AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series
Problem 6 AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series
Problem 7 AMC 8 Daily Practice - The Sum of a Finite Arithmetic Series
Think Academy Powered by ChatGPT