Exercise 1

The second term of a geometric sequence is 6, and the fifth term is 48. Determine the sequence.

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Exercise 2

The 1st term of a geometric sequence is 3 and the eighth term is 384. Find the common ratio, the sum, and the product of the first 8 terms.

Exercise 3

Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

Exercise 4

Calculate the sum of the terms of the following geometric sequence:

1, \frac { 1 }{ 2 }, \frac { 1 }{ 4 }, \frac { 1 }{ 8 }, \frac { 1 }{ 16 }, ...

Exercise 5

Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

Exercise 6

John has purchased 20 books. The 1st book costs 1 dollar, the 2nd, 2 dollars, the 3rd, 4 dollars, and the 4th, 8 dollars, and so on. How much did John pay for the 20 books?

Exercise 7

The sides of a square, l, have lines drawn between them connecting adjoining sides with their midpoints. This creates another square within the original and this process is continued indefinitely. Calculate the sum of the areas of the infinite squares.

Exercise 8

Calculate the fraction that is equivalent to 0.18181818...

Exercise 9

Calculate the fraction that is equivalent to 3.2777777...

 

 

Solution of exercise 1

The second term of a geometric sequence is 6, and the fifth term is 48. Determine the sequence.

{ a }_{ 2 } = 6 \qquad { a }_{ 5 } = 48

{ a }_{ n } = { a }_{ k } . { r }^{ n - k }

48 = 6 { r }^{ 5 - 2 }

{ r }^{ 3 } = 8

r = 2

{ a }_{ 1 } = \frac { { a }_{ 2 } }{ r } = \frac { 6 }{ 2 } = 3

3, 6, 12, 24, 48, ...

 

Solution of exercise 2

The 1st term of a geometric sequence is 3 and the eighth term is 384. Find the common ratio, the sum, and the product of the first 8 terms.

{ a }_{ 1 } = 3 \qquad { a }_{ 8 } = 384

r = \frac { { a }_{ n } }{ { a }_{ n - 1 } } \qquad { S }_{ n } = \frac { { a }_{ n } r - { a }_{ 1 } }{ r - 1 } \qquad P = \pm \sqrt { { ({ a }_{ 1 } . { a }_{ n } ) }^{ n } }

384 = 3 . { r }^{ 8 - 1 }; \qquad { r }^{ 7 } = 128 \qquad { r }^{ 7 } = { 2 }^{ 7 }; \qquad r = 2

{ S }_{ 8 } = \frac { (384 . 2 - 3) }{ (2 - 1) } = 765

{ P }_{ 8 } = \sqrt { { ( 3 . 384 ) }^{ 8 } } = 1, 761, 205, 026, 816

 

Solution of exercise 3

Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

{ S }_{ 5 } = \frac { 48 . 2 - 3 }{ 2 - 1 } = 93

 

Solution of exercise 4

Calculate the sum of the terms of the following geometric sequence:

1, \frac { 1 }{ 2 }, \frac { 1 }{ 4 }, \frac { 1 }{ 8 }, \frac { 1 }{ 16 }, ...

S = \frac { 1 }{ 1 - \frac { 1 }{ 2 } } = \frac { 1 }{ \frac { 1 }{ 2 } } = 2

 

Solution of exercise 5

Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

{ P }_{ 5 } = \sqrt { { (3 . 48) }^{ 5 } } = \sqrt { { (3 . 3 . { 2 }^{ 4 } ) }^{ 5 } } = \sqrt { { 3 }^{ 10 } . { 2 }^{ 20 } } = { 3 }^{ 5 } . { 2 }^{ 10 } = 248 832

 

Solution of exercise 6

John has purchased 20 books. The 1st book costs 1 dollar, the 2nd, 2 dollars, the 3rd, 4 dollars, and the 4th, 8 dollars, and so on. How much did John pay for the 20 books?

{ a }_{ 1 } = 1 \qquad r = 2 \qquad n = 20

{ S }_{ n } = \frac { { a }_{ n } . r - { a }_{ 1 } }{ r - 1 }

S = \frac { (1 . { 2 }^{ 20 - 1 } - 1) }{ (2 - 1) } = 1, 048, 575 dollars

 

Solution of exercise 7

The sides of a square, l, have lines drawn between them connecting adjoining sides with their midpoints. This creates another square within the original and this process is continued indefinitely. Calculate the sum of the areas of the infinite squares.

1 , \frac { 1 }{ \sqrt { 2 }}, \frac { 1 }{ 2 }, \frac { 1 }{ 2 \sqrt { 2 }}, ...

{ 1 }^{ 2 }, \frac { { 1 }^{ 2 } }{ 2 }, \frac { { 1 }^{ 2 } }{ 4 }, \frac { { 1 }^{ 2 } }{ 8 }, ...

S = \frac { { I }^{ 2 } }{ 1 - \frac { 1 }{ 2 } } = \frac { { I }^{ 2 } }{ \frac { 1 }{ 2 } } = 2 { I }^{ 2 }

 

Solution of exercise 8

Calculate the fraction that is equivalent to 0.18181818...

0.18181818... = 0.18 + 0.0018 + 0.000018 + ...

S = \frac { { a }_{ 1 } }{ 1 - r }

S = \frac { 0.18 }{ (1 - 0.01) } = \frac { 2 }{ 11 }

 

Solution of exercise 9

Calculate the fraction that is equivalent to 3.2777777...

3.2777777... = 3.2 + 0.07 + 0.007 + 0.0007 + ...

{ a }_{ 1 } = 0.07 \qquad r = 0.1

3.2 + \frac { 0.07 }{ (1 - 0.1) } = \frac { 32 }{ 10 } + \frac { 7 }{ 90 } = \frac { 59 }{ 18 }

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.