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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.

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

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing rational function IV

Graphing 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

Geometric Sequence

x- and y-Intercepts

Bounded Sequence

Graphing rational function 8

L’Hôpital’s Rule

Graphing polynomial function I

Graphing exponential function I

Absolute and Relative Maxima and Minima

Maxima and Minima

Types of Functions

Asymptotes

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Inverse Function

Graphing rational function II

Graphing logarithmic function

Graphing polynomial function II

Quadratic Function

Absolute Value Function

Rational Functions

Sequences

Bounded Functions

Logarithmic Functions

Properties of Limits

Formulas

Sequence Formulas

Exercises

Inflection Points Worksheet

Maxima and Minima Worksheet