The first term of a geometric progression is 1 and the second term is 2 cos x , where 0 < x < π 2 . Find the set of values of x for which this progression is convergent.

Let a be the first term and r be the common ratio of the geometric progression. Given that, the first term is 1 i.e., a = 1 and second term is 2 cos x i.e, a r = 2 cos x ⇒ r = 2 cos x [Using a = 1 ] We know that, an infinite geometric series converges only when - 1 < r < 1 Therefore, - 1 < 2 cos x < 1 ⇒ - 1 2 < cos x < 1 2 ⇒ 0 < cos x < 1 2 [Since 0 < x < π 2 ] ⇒ π 2 < x < π 3 Hence, the geometric progression is convergent when π 3 < x < π 2 .

The first term of a geometric progression is 1 and the second term is 2 cos x , where 0 < x < π 2 . Find the set of values of x for which this progression is convergent.

Let a be the first term and r be the common ratio of the geometric progression. Given that, the first term is 1 i.e., a = 1 and second term is 2 cos x i.e, a r = 2 cos x ⇒ r = 2 cos x [Using a = 1 ] We know that, an infinite geometric series converges only when - 1 < r < 1 Therefore, - 1 < 2 cos x < 1 ⇒ - 1 2 < cos x < 1 2 ⇒ 0 < cos x < 1 2 [Since 0 < x < π 2 ] ⇒ π 2 < x < π 3 Hence, the geometric progression is convergent when π 3 < x < π 2 .

The first term of a geometric progression is 1and the second term is 2cosx where 0x2 Find the set of values of xfor which this progression is convergent

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6E>Q 16

A circle of radius $1$ cm is drawn touching the three edges of an equilateral triangle.Three smaller circles are then drawn at each corner to touch the original circle and two edges of the triangle.This process is then repeated an infinite number of times, as shown in the diagram. Find the sum of the circumferences of all the circles.

Question Image

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6E>Q 16

A circle of radius $1$ cm is drawn touching the three edges of an equilateral triangle.Three smaller circles are then drawn at each corner to touch the original circle and two edges of the triangle.This process is then repeated an infinite number of times, as shown in the diagram. Find the sum of the areas of all the circles.

Question Image

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 1

The first term of geometric progression is $16$ and the second term is $24$ . Find the sum of the first eight terms geometric progression.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 2

The first term of a geometric progression is $20$ and second term is $16$ . Find the sum to infinity.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 3

The first, second and third terms of a geometric progression are the first, fourth and tenth terms, respectively, of an arithmetic progression. Given that the first term in each progression is $12$ and the common ratio of the geometric progression is $r$ , where $r \neq 1$ , find the value of $r$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 3

The first, second and third terms of a geometric progression are the first, fourth and tenth terms, respectively, of an arithmetic progression. Given that the first term in each progression is $12$ and the common ratio of the geometric progression is $r$ , where $r \neq 1$ , find the sixth term of each progression.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 5

The first, second and third terms of a geometric progression are the first, sixth and ninth terms, respectively, of an arithmetic progression. Given that the first term in each progression is $100$ and the common ratio of the geometric progression is $r$ , where $r \neq 1$ , find the value of $r$ .[Write your answer as decimal]

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 6 - Series>EXERCISE 6F>Q 5

The first, second and third terms of a geometric progression are the first, sixth and ninth terms, respectively, of an arithmetic progression. Given that the first term in each progression is $100$ and the common ratio of the geometric progression is $r$ , where $r \neq 1$ , find the fifth term of each progression.

The first term of a geometric progression is $1$ and the second term is $2 \cos x$ , where $0 < x < \frac{π}{2}$ . Find the set of values of $x$ for which this progression is convergent.

Important Questions on Series

Learn and Prepare for any exam you want !

The first term of a geometric progression is 1 and the second term is 2cosx, where 0<x<π2. Find the set of values of x for which this progression is convergent.

Important Questions on Series

Learn and Prepare for any exam you want !

The first term of a geometric progression is $1$ and the second term is $2 \cos x$ , where $0 < x < \frac{π}{2}$ . Find the set of values of $x$ for which this progression is convergent.