Exercise 1

The fourth term of an arithmetic sequence is 10 and the sixth term is 16. Determine the sequence.

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Exercise 2

The first term of an arithmetic sequence is -1 and the fifteenth term is 27. Find the common difference and the sum of the first fifteen terms.

Exercise 3

Find the sum of the first fifteen multiples of 5.

Exercise 4

Find the sum of the first fifteen numbers ending in5.

Exercise 5

Find the sum of the first fifteen even numbers greater than 5.

Exercise 6

Find the angles of a convex quadrilateral, knowing they are in arithmetic sequence and d = { 25 }^{ \circ }.

Exercise 7

The lower leg of a right triangle is 8 cm in length. Calculate the other two knowing that the sides of the triangle form an arithmetic sequence.

Exercise 8

Calculate three numbers in an arithmetic sequence, whose sum is 27 and the sum of their squares is \frac { 511 }{ 2 }.

 

 

Solution of exercise 1

The fourth term of an arithmetic sequence is 10 and the sixth term is 16. Determine the sequence.

{ a }_{ 4 } = 10 \qquad { a }_{ 6 } = 16

{ a }_{ n } = { a }_{ k } + (n - 1) . d

16 = 10 + (9 - 4) d

d = 3

 

{ a }_{ 1 } = { a }_{ 4 } - 3d

{ a }_{ 1 } = 10 - 9 = 1

Sequence = 1, 4, 7, 10, 13, ...

 

Solution of exercise 2

The first term of an arithmetic sequence is -1 and the fifteenth term is 27. Find the common difference and the sum of the first fifteen terms.

{ a }_{ 1 } = -1 \qquad { a }_{ 15 } = 27

{ a }_{ n } = { a }_{ 1 } + (n - 1) . d

27 = -1 + (15 - 1) d

d = 2

 

{ S }_{ n } = \frac { n }{ 2 } ({ a }_{ 1 } + { a }_{ n })

S = \frac { 15 }{ 2 } (-1 + 27)

S = 195

 

Solution of exercise 3

Find the sum of the first fifteen multiples of 5.

{ a }_{ 5 } = 5 \qquad d = 5 \qquad n = 15

{ a }_{ n } = { a }_{ 1 } + (n - 1) . d

{ a }_{ 15 } = 5 + (15 - 1) 5

{ a }_{ 15 } = 75

 

{ S }_{ n } = \frac { n }{ 2 } ({ a }_{ 1 } + { a }_{ n })

S = \frac { 15 }{ 2 } (5 + 75)

S = 600

 

Solution of exercise 4

Find the sum of the first fifteen numbers ending in 5.

{ a }_{ 1 } = 5 \qquad d = 10 \qquad n = 15

{ a }_{ n } = { a }_{ 1 } + (n - 1) . d

{ a }_{ 15 } = 5 + (15 - 1) 10

{ a }_{ 15 } = 145

 

{ S }_{ n } = \frac { n }{ 2 } ({ a }_{ 1 } + { a }_{ n })

S = \frac { 15 }{ 2 } (5 + 145)

S = 1125

 

Solution of exercise 5

Find the sum of the first fifteen even numbers greater than 5.

{ a }_{ 1 } = 6 \qquad d = 2 \qquad n = 15

{ a }_{ n } = { a }_{ 1 } + (n - 1) . d

{ a }_{ 15 } = 6 + (15 - 1) 2

{ a }_{ 15 } = 34

 

{ S }_{ n } = \frac { n }{ 2 } ({ a }_{ 1 } + { a }_{ n })

S = \frac { 15 }{ 2 } (6 + 34)

S = 1125

 

Solution of exercise 6

Find the angles of a convex quadrilateral, knowing they are in arithmetic sequence and d = { 25 }^{ \circ }.

The sum of the interior angles of a quadrilateral is { 360 }^{ \circ }.

360 = ({ a }_{ 1 } + { a }_{ 4 } ) . \frac { 4 }{ 2 }

{ a }_{ 4 } = { a }_{ 1 } + 3 . 25

 

{ S }_{ n } = \frac { n }{ 2 } ({ a }_{ 1 } + { a }_{ n })

360 = \frac { 4 }{ 2 } ({ a }_{ 1 } + { a }_{ 1 } + 3 \times 25)

{ a }_{ 1 } = 52.5

Hence,

{ a }_{ 1 } = 52.5 \qquad { a }_{ 2 } = 77.5 \qquad { a }_{ 3 } = 102.5 \qquad { a }_{ 4 } = 127.5

 

Solution of exercise 7

The lower leg of a right triangle is 8 cm in length. Calculate the other two knowing that the sides of the triangle form an arithmetic sequence.

{ a }_{ 2 } = 8 + d \qquad { a }_{ 3 } = 8 + 2d

{ (8 + 2d) }^{ 2 } = { (8 + d) }^{ 2 } + 64

64 + 32d + 4 { d }^{ 2 } = 64 + 16d + { d }^{ 2 } + 64

3 { d }^{ 2 } + 16d - 64 = 0

d = \frac { 8 }{ 3 } \qquad d = -8

d = -8 is rejected because length is always positive.

8, \frac { 32 }{ 3 }, \frac { 40 }{ 3 }

 

Solution of exercise 8

Calculate three numbers in an arithmetic sequence, whose sum is 27 and the sum of their squares is \frac { 511 }{ 2 }.

Central term x

  1. x - d
  2. x
  3. x + d

 

x - d + x + x + d = 27

x = 9

 

{ (9 - d) }^{ 2 } + 81 + { (9 + d) }^{ 2 } = \frac { 511 }{ 2 }

d = \pm \frac { 5 }{ 2 }

\frac { 13 }{ 2 }, 9, \frac { 23 }{ 2 }

\frac { 23 }{ 2 }, 9, \frac { 13 }{ 2 }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.