Download Geometric Sequences and Series: Definition, Common Ratio, Nth Term, and Sum and more Exams Analytical Geometry and Calculus in PDF only on Docsity!
A geometric sequence (or progression) may be defined as:
{ n }
i i
A sequence a where each pair of consecutive terms has the
same nonzero ratio, r a , r .The number r is called the common a ratio of the sequence.
1
โ
Note that the definition will give the recursive formula a i+1 = rai. The following example will show how to find the common ratio of a geometric sequence.
Example 1 : Find the common ratio of the geometric sequence an^2 n 3
Solution:
Step 1: Substitute into the ratio formula.
The sequence is substituted into the definition.
i^ i i (^) i
r a a (^) (^1 )
Step 2: Solve for r.
( )
i i i i i i i i
r
r
r
1 1 1 1
โ โ โ โ โ
= โ^ โโ^ โ= =
The following is a definition for nth term of a geometric sequence.
The nth term of a geometric sequence, whose first term is a 1 and whose common ratio is r, is given by the formula an = a 1 r n โ 1^.
The three examples following will show various means to find the nth term of geometric sequences.
Geometric Sequences
Example 2 : Find the first five terms of the geometric sequence whose first term is a 1 = 3 and whose common ratio is r = โ2.
Solution:
Step 1: Substitute the given information into the definition and solve.
a Given a a a a
1 2 1 1 2 3 1 2 3 4 1 3 4 5 1 4 5
โ = โ = โ = โ =
Example 3 : Find the twelfth term of the geometric sequence 5, -15 , 45 , โฆ.
Solution:
Step 1: Analysis.
The terms given are a 1 = 5 and n = 12. The common ratio, using the definition given is r^15 5
Step 2: Substitute and solve.
n an a r a a , a ,
1 1 12 1 11 12 12 12
โ โ =
Example 4 (Continued):
Step 4: Substitution.
The value of r found in step 3 is substituted back into the first equation to solve for a 1
a r
a
a a
1 3 3 1 3 1 1
Step 5: Substitute and solve for a 14.
an a r^ n
a
a
1 1 14 1 13 14
14
โ โ =
= โ^ โ
To find the sum of a geometric series, either of the following two formulas may be used:
n Sn a^ a r or Sn a^ ran r r
= โ =^ โ
; r โ
Example 5 will show how to find the sum of a geometric series.
Example 5 : Find the sum of the first seven terms of the geometric series 2 + 6 + 18 + โฆ.
Solution:
Step 1: Analysis.
The terms a 1 = 2 and n = 7 are given. The common ratio is found to be r^6 2
Example 5 (Continued):
Step 2: Substitute and solve.
n n S a^ a r r
S
1 1
7 7
โ โกโฃ^ โคโฆ
= โ^ =โ
The final topic to be covered is that concerning the sum of an infinite series. The definition of the sum of an infinite geometric series is:
If r , then the infinite geometric series has the sum S a r
The final example shows its use.
Example 6 : Find the sum of the infinite geometric series whose first term is 4 and whose common ratio is -0.6.
Solution:
Step 1: Substitute and solve.
S a r S....
1 1 4 1 0 6 4 1 0 6 4 1 6 2 5
