Page 721 - Algebra 1

Example
2
Extending Geometric Sequences
Find the next four terms in the geometric sequence.
a. 2, 8, 32, 128, ...
SOLUTION
The common ratio is 4. Each term of the sequence is 4 times the previous term. Use the common ratio to find the next 4 terms.
×4×4×4×4
128 512 2048 8192 32,768
The next 4 terms of the sequence are 512, 2048, 8192, and 32,768.
b. 250, -50, 10, -2, ...
SOLUTION
The common ratio is -_1 . Use the common ratio to find the next 4 terms. 5
× ( - _1 ) × ( - _1 ) × ( - _1 ) × ( - _1 ) 5555
-2 _2 -_2 _2 -_2 5 25 125 625
The next 4 terms of the sequence are _2, -_2 , _2 , and -_2 . 525125 625
Examine the terms of the sequence in Example 2a.
Term 1: 2 = 2 · 1 = 2 · 40 Term 2: 8 = 2 · 4 = 2 · 41
Term 3: 32 = 2 · 4 · 4 = 2 · 42 Term 4: 128 = 2 · 4 · 4 · 4 = 2 · 43
The exponent on the common ratio 4 is 1 less than the number of the term. In general, the nth term of the sequence is 2 · 4n-1.
Finding the nth Term of a Geometric Sequence a. The first term of a geometric sequence is 7 and the common ratio
is -3. Find the 6th term in the sequence. SOLUTION
Math Reasoning
Generalize When will the common ratio be negative?
Math Reasoning
Analyze Which
operation is equivalent
to multiplying by -_1 ? 5
Finding the nth Term of a Geometric Sequence
Let A(n) equal the nth term of a geometric sequence, then A (n ) = a r n - 1
where a is the first term of the sequence and r is the common ratio.
Reading Math
In the expression 4n-1, n represents the integers 1, 2, 3, 4, and so on.
Example
3
706 Saxon Algebra 1
A(n) = ar n - 1 A(6) = 7(-3) 6-1
= 7(-3)5 = 7(-243) = -1701
Use the formula.
Substitute 6 for n, 7 for a, and -3 for r. Simplify the exponent.
Raise -3 to the 5th power.
Multiply.
The 6th term in the sequence is -1701.