8.3 Geometric Sequences And Series
A sequence is geometric when the ratios of consecutive terms are the same. The sequence a1, a2, a3, a4, ..., an, is geometric when there is a number r such that:
a2/a1 = a3/a2 = a4/a3 = ... = r, r ≠ 0
The number r is the common ratio of the sequence.
a2/a1 = a3/a2 = a4/a3 = ... = r, r ≠ 0
The number r is the common ratio of the sequence.
The sequence
1, 4, 9, 16, ...
whose nth term is n², is not geometric. The ratio of the second term to the first term is
a2/a1 = 4/1 = 4
but the ratio of the third term to the second term is
a3/a2 = 9/4
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1rⁿ ⁻ ¹
where r is the common ratio of consecutive terms of the sequence. Every geometric sequence can be written in the following form,
a1, a1r, a1r², a1r³, a1r⁴, ..., a1rⁿ ⁻ ¹, ...
The (n + 1)th term of a sequence can be found by multiplying the nth term by r.
an + 1 = anr
1, 4, 9, 16, ...
whose nth term is n², is not geometric. The ratio of the second term to the first term is
a2/a1 = 4/1 = 4
but the ratio of the third term to the second term is
a3/a2 = 9/4
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1rⁿ ⁻ ¹
where r is the common ratio of consecutive terms of the sequence. Every geometric sequence can be written in the following form,
a1, a1r, a1r², a1r³, a1r⁴, ..., a1rⁿ ⁻ ¹, ...
The (n + 1)th term of a sequence can be found by multiplying the nth term by r.
an + 1 = anr
Finding the Terms of a Geometric Sequence
Finding a Term of a Geometric Sequence
With any two terms of a geometric sequence, the information can be used to find a formula for the nth term of the sequence.
The Sum of a Finite Geometric Sequence
The sum of the finite geometric sequence
a1, a1r, a1r², a1r³, a1r⁴, ..., a1rⁿ ⁻ ¹
with common ratio r ≠ 1 is given by
The sum of the finite geometric sequence
a1, a1r, a1r², a1r³, a1r⁴, ..., a1rⁿ ⁻ ¹
with common ratio r ≠ 1 is given by
Finding the Sum of a Finite Geometric Sequence
When using the formula for the sum of a geometric sequence, be careful to check that the index begins at i = 1. For an index that begins at i = -, you must adjust the formula for the nth partial sum.
Geometric Series
The sum of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series. the formula for the sum of a finite geometric sequence, can depending on the value of r, be extended to produce a formula for the sum of an infinite geometric series. If the common ratio r has the property that
l r l < 1
then it can be shown that rⁿ becomes arbitrarily close to zero as n increases without bond. Consequently.
a1((1 - rⁿ)/(1 - r)) -> a1((1 - 0)/(1 - r)) as n ->∞
The sum of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series. the formula for the sum of a finite geometric sequence, can depending on the value of r, be extended to produce a formula for the sum of an infinite geometric series. If the common ratio r has the property that
l r l < 1
then it can be shown that rⁿ becomes arbitrarily close to zero as n increases without bond. Consequently.
a1((1 - rⁿ)/(1 - r)) -> a1((1 - 0)/(1 - r)) as n ->∞
Finding the Sum of an Infinite Geometric Series
Application