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Hamza

March 17, 2021

Chapters

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6
Solution of exercise 7
Solution of exercise 8

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

$x - d$
$x$
$x + d$

$x - d + x + x + d = 27$

$x = 9$

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

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

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

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

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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.

Comments

Arithmetic Sequence Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Solution of exercise 8

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Theory

Exponent Rules

Absolute Value Function

Asymptotes

Bounded Functions

Bounded Sequence

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Coordinates in the Plane

Even and Odd Functions

Exponential Function

Graphing polynomial function I

Graphing logarithmic function

Graphing polynomial function II

Graphing rational function II

Graphing rational function IV

Graphing rational function V

Graphing rational function 8

Graphing exponential function I

Geometric Sequence

Graphing Functions

x- and y-Intercepts

Inverse Function

L’Hôpital’s Rule

Limit of a Sequence

Logarithmic Functions

Absolute and Relative Maxima and Minima

Maxima and Minima

nth Term

Optimization

Properties of Limits

Quadratic Function

Rational Functions

Sequences

Types of Functions

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Concave and Convex Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Geometric Sequence Problems

Piecewise Functions

Trigonometric Functions

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Graphing exponential function II

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Formulas

Sequence Formulas

Exercises

Absolute Value Functions Worksheet