Learn – Math circle – Sequences

1 Lesson notes

  1. An arithmetic sequence or an arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers in the sequence is the same. Examples of arithmetic sequences include:
    1. 2,3,5,8,\cdots
    2. 10, 1, -8, -17, -26, \cdots
    3. -3,-4.5,-6,-7.5, \cdots
  2. A geometric sequence or a geometric progression is a sequence of numbers such that the ratio between any two consecutive numbers in the sequence is the same. Examples of geometric sequences include:
    1. 3,6,12,24,48\cdots
    2. 10, 0.1, 0.01, 0.001, \cdots

2 Finding terms

2.1 Exercises

2.1.1 Problem set

For each of the following problems, find the required term of the given sequence.

  1. 11\textsuperscript{th} term of 7, 11, 15, 19,\cdots
  2. 100\textsuperscript{th} term of 2, 1.9, 1.8, 1.7, 1.6, 1.5,\cdots
  3. 10\textsuperscript{th} term of 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \cdots
  4. 15\textsuperscript{th} term of 2, \frac{3}{2}, \frac{4}{4}, \frac{5}{8}, \frac{6}{16},\cdots

2.1.2 Problem set

  1. The 10\textsuperscript{th} and 15\textsuperscript{th} terms of an arithmetic sequence are 13 and 28 respectively.
    1. Find the 19\textsuperscript{th} term of the sequence.
    2. Find the 4\textsuperscript{th} term of the sequence.
  2. The 9\textsuperscript{th} and 17\textsuperscript{th} terms of an arithmetic sequence are 23 and 19 respectively.
    1. Find the 21\textsuperscript{st} term of the sequence.
    2. Find the 4\textsuperscript{th} term of the sequence.
  3. The 3\textsuperscript{rd} and 6\textsuperscript{th} terms of a geometric sequence are 6 and 162 respectively.
    1. Find the 8\textsuperscript{th} term of the sequence.
    2. Find the first term of the sequence.
  4. The 5\textsuperscript{th} and 9\textsuperscript{th} terms of a geometric sequence are 96 and 6 respectively.
    1. Find the 11\textsuperscript{th} term of the sequence.
    2. Find the 3\textsuperscript{rd} term of the sequence.
  5. The 200\textsuperscript{th} and the 300\textsuperscript{th} terms of a geometric sequence are 20 and 30 respectively. What is the 400\textsuperscript{th} term of the sequence?

3 Finding sums

3.1 Exercises

3.1.1 Problem set

For each of the following problems, find the sum of the required number of terms of the given sequence.

  1. 24 terms of 8, 12, 17, \cdots
  2. 100 terms of 10, 9.9, 9.8, \cdots
  3. 8 terms of 3, 6, 12, \cdots
  4. 7 terms of 162, 54, 18, \cdots

3.1.2 Problem set

  1. The 14\textsuperscript{th} and 34\textsuperscript{th} terms of an arithmetic sequence are 18 and 48 respectively.
    1. Find the sum of the sequence from its 14\textsuperscript{th} term through its 34\textsuperscript{th} term.
    2. Find the sum of the sequence from its 10\textsuperscript{th} term through its 40\textsuperscript{th} term.
  2. The sum of an arithmetic sequence from its 300\textsuperscript{th} term through its 399\textsuperscript{th} term is 67.89. The common difference d for the sequence is 0.00345. What is the sum of the arithmetic sequence from its 100\textsuperscript{th} term through its 199\textsuperscript{th} term?
  3. 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8},\cdots is a geometric sequence. If you add the terms of the sequence starting from the very first term, would the sum ever become bigger than 2? How many terms must you add to get the sum to be bigger than 1.9999? (You are allowed to use calculator for this problem)

3.1.3 Problem set

  1. a, a+d, a+2d, a+3d, \cdots and b, b+e, b+2e, b+3e, \cdots are two arithmetic sequences. A new sequence is formed by adding the two sequences term by term. That is, the first term of the new sequence is obtained by adding the first terms of the two given sequences, the second term of the new sequence is obtained by adding the second terms of the two given sequences and so on. What is the sum of the first n terms of the new sequence?
  2. a, ar, ar^2, ar^3, \cdots and b, bs, bs^2, bs^3, \cdots are two geometric sequences. A new sequence is formed by dividing the two sequences term by term. That is, the first term of the new sequence is obtained by dividing the first term of the first sequence by the first term of the second sequence, the second term of the new sequence is obtained by dividing the second term of the first sequence by the second term of the second sequence, and so on. What is the sum of the first n terms of the new sequence?