Identify which of the following are geometric sequences.
For those that are, write down the first term and the common ratio.
(i)
(ii)
(iii)
(iv)
Did this page help you?
Exam code: 9MA0
Identify which of the following are geometric sequences.
For those that are, write down the first term and the common ratio.
(i)
(ii)
(iii)
(iv)
How did you do?
Did this page help you?
A geometric sequence has a first term, , a common ratio,
, and an
th term,
, where
.
Find a formula for for each of the following geometric sequences:
(i)
(ii) ,
(iii) ,
How did you do?
Did this page help you?
Find the 5th and 10th terms of each of the following geometric sequences:
(i)
(ii) giving your answers to 2 decimal places
(iii)
How did you do?
Did this page help you?
A geometric series has a first term of 5 and a common ratio of .
Use the formula to find the sum of the first 12 terms, to the nearest whole number.
How did you do?
A different geometric series has a first term of 4 and a common ratio of .
Use the formula to find the sum to infinity.
How did you do?
Did this page help you?
The first term of a geometric series is 6.
The sum to infinity of the series is 8.
Show that the common ratio is 0.25.
How did you do?
Explain why the sum to infinity exists for the geometric series in part (a).
How did you do?
Did this page help you?
For a geometric sequence,
the first term is 900
the common ratio is where
the 18th term is 18
Show that satisfies the equation
How did you do?
Find the value of correct to 3 significant figures.
How did you do?
Did this page help you?
In this question you should show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
A company made a profit of £20 000 in its first year of trading, Year 1.
A model for future trading predicts that the yearly profit will increase by 8% each year, so that the yearly profits will form a geometric sequence.
According to the model, show that the profit for Year 3 will be £23 328.
How did you do?
According to the model, find the first year when the yearly profit will exceed £65 000.
How did you do?
According to the model, find the total profit for the first 20 years of trading, giving your answer to the nearest £1 000.
How did you do?
Did this page help you?
A car has six forward gears.
The fastest speed of the car
in 1st gear is 28 km h-1
in 6th gear is 115 km h-1
Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
find the fastest speed of the car in 5th gear.
How did you do?
Did this page help you?
A geometric sequence is defined as follows
The first term is 2
The sixth term is 486
Find the common ratio.
How did you do?
The sum of the first terms is
.
Show that
How did you do?
Hence find the value of .
How did you do?
Did this page help you?
In a geometric sequence
the 3rd term is 10
the 6th term is 270
Find the first term and the common ratio.
How did you do?
In a different geometric sequence, the 12th term is 16 times greater than the 8th term.
Find the possible values of the common ratio.
How did you do?
Did this page help you?
The first three terms of a geometric sequence are given by
where
Show that
How did you do?
Find the value of the 15th term of the sequence.
How did you do?
State, with a reason, whether 8192 is a term in the sequence.
How did you do?
Did this page help you?
In a geometric series,
the first term is 19
the common ratio is
the sum of the first terms of the series is greater than 56
(i) Show that
(ii) Hence find the smallest possible value of .
How did you do?
Did this page help you?
The sum of the first two terms in a geometric series is 9.31
The sum of the first four terms in the same geometric series is 11.02
The common ratio of the geometric series is where
Show that
How did you do?
Hence find the possible values of .
How did you do?
Did this page help you?
The first three terms in a geometric sequence are ,
,
, where
is a constant.
(i) Show that
(ii) Hence find the value of .
How did you do?
Find the common ratio of the sequence.
How did you do?
Find the sum of the first 12 terms.
How did you do?
Did this page help you?
A geometric series is defined as follows:
The first term is 64
The sum to infinity is 384
Show that the common ratio is .
How did you do?
Find the difference between the 9th term and the 10th term of the series, to 3 significant figures.
How did you do?
Find the sum of the first eight terms in the series, to 3 significant figures.
How did you do?
Given that the sum of the first terms of the series is greater than 380, find the smallest possible value of
.
How did you do?
Did this page help you?
Given that the geometric series
is convergent, find the range of possible values of
How did you do?
Assuming the series is convergent, find an expression for the sum to infinity of the series in terms of .
How did you do?
Did this page help you?
The first term of a geometric series is , and its common ratio is 5.
A different geometric series has a first term of and a common ratio of 3.
For both series, the sum of the first three terms are equal.
Find the value of , giving your answer as a fraction in simplest form.
How did you do?
Did this page help you?
A geometric sequence is given by
where is a constant such that
.
Find, in terms of and
, a formula for the
th term of the sequence.
How did you do?
The sequence is part (a) is used to form a series.
Show that the sum to infinity of the series is
How did you do?
Given that the sum to infinity is , find the value of
.
How did you do?
Did this page help you?
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
A geometric series has common ratio and first term
.
Given and
, prove that
How did you do?
Given also that is four times
, find the exact value of
.
How did you do?
Did this page help you?
The first three terms of a geometric sequence are
where is a constant.
Show that satisfies the equation
How did you do?
Given that the sequence converges,
(i) find the value of , giving a reason for your answer,
(ii) find the value of
How did you do?
Did this page help you?
The first three terms of a geometric sequence are given by
where is a non-zero real number.
Find the value of the 102nd term in the sequence.
How did you do?
Did this page help you?
Find the value of
How did you do?
Find the value of
How did you do?
Did this page help you?
In a geometric sequence,
the second term is 4
the common ratio is where
the 16th term is 9
Show that satisfies the equation
How did you do?
Find the value of , to 3 significant figures.
How did you do?
Did this page help you?
A geometric series has first term of 14 and common ratio of .
Given that the sum of the first terms of the series is less than 1000, find the largest possible value of
.
How did you do?
Did this page help you?
The sum of the first three terms in a geometric series is 8.75
The sum of the first six terms in the same geometric series is 13.23
Find the common ratio of the series.
How did you do?
Did this page help you?
A geometric series has first term of and common ratio of
.
Show that the sum of the first ten terms of the series is
where is a positive integer to be found.
How did you do?
Did this page help you?
The first three terms in a geometric sequence are ,
,
, where
is a constant.
Find the value of .
How did you do?
Find the sum of the first 12 terms.
How did you do?
Did this page help you?
In a geometric series,
the second term is 13.44
the fifth term is 5.67
Assuming the series is convergent, find the sum to infinity of the series.
How did you do?
Find the difference between the sum to infinity of the series and the sum of the first 20 terms of the series, to 2 decimal places.
How did you do?
Did this page help you?
In a geometric series
the first term is 9
the sum of the first three terms is 19
the common ratio is where
Show that
How did you do?
Find the possible values of .
How did you do?
Given that the series converges, find the sum to infinity of the series.
How did you do?
Did this page help you?
In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
Given that the first three terms of a geometric series are
and
show that
How did you do?
Given that is an obtuse angle measured in radians, solve the equation in part (a) to find the exact value of
How did you do?
Show that the sum to infinity of the series can be expressed in the form
where is a constant to be found.
How did you do?
Did this page help you?
The first three terms of a geometric sequence are given by
where is a non-zero real number.
Find the 6th term in the sequence, giving your answer as a fraction.
How did you do?
Did this page help you?
In a geometric series,
the second term is 648
the fifth term is 375
the sum of the first terms of the series is greater than 4660
Find the smallest value of .
How did you do?
Did this page help you?
The sum of the first four terms in a geometric series is 27.2
The sum of the first eight terms in the same geometric series is 164.9
Given that the first term is positive, find the common ratio of the series.
How did you do?
Did this page help you?
A geometric series has a first term of and its terms satisfy the relationship
for all .
Given that all the terms in the series are positive, show that the sum of the first twelve terms of the series is
where and
are positive integers to be found.
How did you do?
Did this page help you?
The first three terms in a geometric series are ,
,
, where
is a constant.
Find the possible values of .
How did you do?
Given that the sum to infinity exists, find the sum to infinity of this series.
How did you do?
Did this page help you?
In a convergent geometric series
the second term is
the third term is
where .
Find the range of possible values of .
How did you do?
The sum to infinity of the series is
Find the possible values of .
How did you do?
Did this page help you?
The geometric series is convergent, and the sum to infinity of the series is
. The first term of the series is
, and the common ratio is
.
A different series is formed by squaring all the terms of the series
above.
Show that is also a convergent geometric series.
How did you do?
The sum to infinity of series is
.
Express the ratio in terms of
and
, simplifying your answer as far as possible.
How did you do?
Show that if , then
for all .
Hence describe the relationship between the terms of the two series in the case when .
How did you do?
Did this page help you?