Geometric sequences and series

The first 3 terms of a geometric sequence are

where k is a constant.

Using algebra and making your reasoning clear, prove thatk =

Hence find the sum to infinity of the geometric sequence.

The first three terms of a geometric sequence are

3 k + 4 12 – 3 k k + 16

where k is a constant.

Show that k satisfies the equation

Given that the sequence converges,

find the value of k, giving a reason for your answer,

find the value of S_∞

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 12 cos θ 5 + 2 sin θ and 6 tan θ

show that

Given that θ is an obtuse angle measured in radians,

solve the equation in part (a) to find the exact value of θ

show that the sum to infinity of the series can be expressed in the form

where k is a constant to be found.

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

find the first year when the yearly profit will exceed £65 000

find the total profit for the first 20 years of trading, giving your answer to the nearest £1000

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

find the first year when the yearly profit will exceed £65 000

find the total profit for the first 20 years of trading, giving your answer to the nearest £1000

Show that

Show that

A car has six forward gears. The fastest speed of the car ● in 1^st gear is 28 km h^–1 ● in 6^th gear is 115 km h^–1 Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

find the fastest speed of the car in 3^rd gear. 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 5^th gear.

A sequence of numbers a₁, a₂, a₃, … is defined by

where k is a constant. Given that ● the sequence is a periodic sequence of order 3 ● a₁ = 2

show that

For this sequence explain why k 1

Find the value of

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 r and first term a. Given r 1 and a ≠ 0

prove that

Given also that S₁₀ is four times S₅

find the exact value of r.

A competitor is running a 20 kilometre race.

She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down.

In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be 5% greater than the time that she took to complete the previous kilometre.

Using the model,

show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,

estimate the total time, in minutes and seconds, that she will take to complete the race.

Find the value of

Show that

A sequence u₁, u₂, u₃, … is defined by

A geometric series with common ratio r= – 0.9 has sum to infinity 10 000 For this series,

find the first term,

find the fifth term,

find the sum of the first twelve terms, giving this answer to the nearest integer.

The first three terms of a geometric sequence are 7 k – 5, 5 k – 7, 2 k + 10 where k is a constant.

Show that 11k² – 130k + 99 = 0

Given that k is not an integer,

For this value of k,

evaluate the fourth term of the sequence, giving your answer as an exact fraction,

evaluate the sum of the first ten terms of the sequence.

A geometric series has first term a and common ratio r = The sum of the first 4 terms of this series is 175

Show that a = 64

Find the sum to infinity of the series.

Find the difference between the 9th and 10th terms of the series. Give your answer to 3 decimal places.

The first term of a geometric series is 20 and the common ratio is The sum to infinity of the series is S_’

Find the value of S_’

The sum to N terms of the series is S_N

Find, to 1 decimal place, the value of S₁₂

Find the smallest value of N, for which

The first three terms of a geometric series are 18, 12 and p respectively, where p is a constant. Find

the value of the common ratio of the series,

the value of p,

the sum of the first 15 terms of the series, giving your answer to 3 decimal places.

The first three terms of a geometric series are 4 p, (3 p + 15) and (5 p + 20) respectively, where p is a positive constant.

Show that 11p² – 10p – 225 = 0

Hence show that p = 5

Find the common ratio of this series.

Find the sum of the first ten terms of the series, giving your answer to the nearest integer.

A company predicts a yearly profit of £120 000 in the year 2013. The company predicts that the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequence with common ratio 1.05

Show that the predicted profit in the year 2016 is £138 915

Find the first year in which the yearly predicted profit exceeds £200 000

Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.

A geometric series is a + ar + ar² +...

Prove that the sum of the first n terms of this series is given by

The third and fifth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. For this series find,

the common ratio,

the first term,

the sum to infinity.

The second and third terms of a geometric series are 192 and 144 respectively. For this series, find

the common ratio,

the first term,

the sum to infinity,

the smallest value of n for which the sum of the first n terms of the series exceeds 1000.

The third term of a geometric sequence is 324 and the sixth term is 96

Show that the common ratio of the sequence is

Find the first term of the sequence.

Find the sum of the first 15 terms of the sequence.

Find the sum to infinity of the sequence.

A geometric series has first term 5 and common ratio . Calculate

the 20th term of the series, to 3 decimal places,

the sum to infinity of the series.

Given that the sum to k terms of the series is greater than 24.95,

find the smallest possible value of k.