Algebra Progress
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Polynomials
Sequence and Series
Miscellaneous
Geometric Series
Going beyond the school level for geometric series and its appearances in contests.
Author: Jae Gwan Park
Geometric sequences can also be written through a recursive formula, where the th term is a product of the th term and the common ratio. In recursive form, the first term must be defined.
Comparing the explicit (general) and recursive forms:
Explicit:
Recursive:
The sum of an infinite geometric series with is
Let and be real numbers with and such that
and
Find .
The geometric mean is the middle number , when , , and form a geometric sequence in that order.
Consequently we can derive:
In fact, we can extend this theorem. Since all terms are evenly spaced by a ratio ,
only if
Find integer(s) so that and form a geometric sequence.
Here are some harder problems.
(2014 CEMC Euclid Q6b)
The geometric sequence with terms , has .
Also, the product of all terms equals . Determine the values of .
A geometric sequence has terms.
The sum of its first two terms is .
The sum of its first three terms is .
The sum of its first four terms is .
Determine how many of the terms in the sequence are integers.
The first three terms of a geometric progression are and . What is the fourth term?
Harder problems often incorporate other math topics like logarithms, systems of equations, etc.
If , and are consecutive terms of a geometric sequence, determine the possible values of .
The sum of an infinite geometric series is a positive number , and the second term in the series is . What is the smallest possible value of
A sequence of three real numbers forms an arithmetic progression with a first term of . If is added to the second term and is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?