# Introduction to Geometric Sequences and Series

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Introduction to Geometric Sequences and Series
23 May 2011

Investigation: Find the next 3 terms of each sequence:
{3, 6, 12, 24, …} {32, 16, 8, 4, …}

Geometric Sequences Sequences that increase or decrease by multiplying the previous term by a fixed number This fixed number is called r or the common ratio

Finding the Common Ratio
Divide any term by its previous term Find r, the common ratio: {3, 9, 27, 81, …}

Your Turn: Find r, the common ratio: {0.0625, 0.25, 1, 4, …}
{-252, 126, -63, 31.5, …}

Arithmetic vs. Geometric Sequences
Arithmetic Sequences Increases by the common difference d Addition or Subtraction d = un – un–1 Geometric Sequences Increases by the common ratio r Multiplication or Division

Determine if each sequence is arithmetic, geometric, or neither: {2, 7, 12, 17, 22, …} {-6, -3.7, -1.4, 9, …} {-1, -0.5, 0, 0.5, …} {2, 6, 18, 54, 162, …}

Recursive Form of a Geometric Sequence
un = run–1 n ≥ 2 nth term n–1th term common ratio

Example #1 u1 = 2, u2 = 8 Write the recursive formula
Find the next two terms

Example #2 u1 = 14, u2 = 39 Write the recursive formula
Find the next two terms

Your Turn: For the following problems, write the recursive formula and find the next two terms: u1 = 4, u2 = 4.25 u1 = 90, u2 = -94.5

Explicit Form of a Geometric Sequence
common ratio un = u1rn–1 n ≥ 1 nth term 1st term

Example #1 u1 = 2, Write the explicit formula
Find the next three terms Find u12

Example #2 u1 = 6, u2 = 18 Write the explicit formula
Find the next three terms Find u12

Your Turn: For the following problems, write the explicit formula, find the next three terms, and find u12 u1 = 5, r = -¼ u1 = 5, u2 = -20 u1 = 144, u2 = 72

Partial Sum of a Geometric Sequence

Partial Sum of a Geometric Sequence
upper bound (ending term) common ratio lower bound (starting term)

Example #1 k = 9, u1 = -1.5, r = -½

Example #2 k = 6, u1 = 1, u2 = 5

Example #3 k = 8,

Your Turn: Find the partial sum: k = 6, u1 = 5, r = ½
k = 7, u1 = 3, u2 = 6 k = 8, u1 = 24, u2 = 6