4.3 Geometric Progressions

The sum of the first three terms in a geometric series is 8.75.

The sum of the first six terms in the same series is 13.23.

Find the common ratio, , of the series.

A geometric series has first term  and common ratio .

Show that the sum of the first ten terms of the series is equal to , where  is a positive integer to be determined.

The first three terms in a geometric series are  ,  ,  , where  is a constant.

Find the value of .

Find the sum of the first 12 terms in this series.

The second and fifth terms of a geometric series are 13.44 and 5.67 respectively.  The series has first term  and common ratio .

By first determining the values of  and , calculate the sum to infinity of the series.

Calculate the difference between the sum to infinity of the series and the sum of the first 20 terms of the series. Give your answer accurate to 2 decimal places.

A geometric progression has first term 9, and the sum of the first three terms of the series is 19.  The common ratio of the series is .

Show that .

Find the two possible values of .

Given that the sum to infinity of the progression exists, find the sum to infinity of the progression.

The th term of a geometric progression is given by .

Calculate, giving your answers as exact values

The sum to infinity of the progression starting with the seventh term.

The sum to infinity of the progression whose th term is given by , where  is defined as above.

The sum of the first two terms in a geometric series is 9.31.

The sum of the first four terms in the same series is 11.02.

The common ratio of the series is .

 Show that .

Hence find the two possible values of .

The first term of a geometric series is , and its common ratio is 5.  A different geometric series has first term  and common ratio 3.  The sum of the first three terms of both series is the same. 

Find the value of  , giving your answer as a fraction in simplest terms.

The first three terms in a geometric series are , , , where  is a constant.

Show that   .

Hence find the value of .

Find the common ratio, , of this series.

 Find the sum of the first 12 terms in this series.

The first three terms of a geometric progression are  ,  and .  Given that the sum to infinity of the progression exists

write down an inequality that the common ratio of the progression must satisfy, and hence find the range of possible values of

find an expression for the sum to infinity of the progression in terms of .

A geometric progression has first term 64, and the sum to infinity of the progression is 384.

Show that the common ratio, , of the progression is   .

Find the difference between the ninth and tenth terms of the progression, giving your answer correct to 3 significant figures.

Calculate the sum of the first eight terms of the progression, giving your answer correct to 3 significant figures.

Given that the sum of the first terms of the progression is greater than 380, show that

The th term of a geometric progression is given by .

Calculate, giving your answers as exact values

The sum of the first nine terms of the progression.

The sum to infinity of the progression starting from the tenth term.

The sum of the first four terms in a geometric series is 27.2, and the sum of the first eight terms in the same series is 164.9.

Given that the first term of the series is positive, find the common ratio, , of the series.

A geometric series has first term , and its terms are connected by the relationship  for all .

Given that all the terms of the series are positive, show that the sum of the first twelve terms of the series may be written in the form

where  and  are positive integers and  is a surd.

The first three terms in a geometric progression  are , , , where  is a constant.

Find the possible values of .

Given that the sum to infinity of the progression exists, find the sum to infinity of the series progression.

The second and third terms of a  geometric progression are  and , where  is a real number not equal to 1 or -1.

Given that the sum to infinity of the progression exists,

Find the range of possible values of .

Given that the sum to infinity of the series is -6,

find the two possible values of  .

The geometric progression is defined by  , where denotes the term of the progression. The sum to infinity of the progression exists and  is denoted by .  The first term of the progression is , and the common ratio is .

A different progression   is formed by squaring all the terms of the progression  above.

Show that  is also a  geometric progression, and that its sum to infinity also exists.

The sum to infinity of the progression T  is .

Express the ratio in terms of  and , simplifying your answer as far as possible.

Show that if , then  for all .  Comment on what this shows about the relationship between the terms of the two progressions.

The th term of a geometric progression is given by

Calculate the sum of the eleventh through twenty-third terms of the sequence whose th term is given by  where  is defined as above.  You should give your answer as an exact value.