# Arithmetic Sequence Problems

### Solutions

1The fourth term of an arithmetic sequence is 10 and the sixth term is 16. Determine the sequence.

2The 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.

3Find the sum of the first fifteen multiples of 5.

4Find the sum of the first fifteen numbers ending in 5.

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

6Find the angles of a convex quadrilateral, knowing they are in arithmetic sequence and d = 25º.

7The 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.

8Calculate three numbers in an arithmetic sequence, whose sum is 27 and the sum of their squares is 511/2.

## 1

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

a_{ 4 }= 10; a_{ 6 }= 16

**a _{ n } = a_{ k } + (n − k) · d**

16 = 10 + (6 − 4) d; d= 3

a_{1}= a_{4} − 3d;

a_{1} = 10 − 9 = 1

1, 4, 7, 10, 13, ...

## 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; a_{ 15 }= 27;

**a _{ n } = a_{ 1 } + (n − 1) · d**

27= −1 + (15 − 1) d; 28 = 14d; d = 2

S= (−1 + 27) 15/2 = 195

## 3

Find the sum of the first fifteen multiples of 5.

a_{1}= 5; d= 5; n = 15.

**a _{ n } = a_{ 1 } + (n − 1) · d**

a_{15} = 5 + 14 · 5 = 75

S_{15} = (5 + 75)· 15/2 = 600.

## 4

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

a_{1}= 5; d= 10 ; n= 15.

a_{15}= 5 + 14 ·10= 145

S_{15} = (5 + 145)· 15/2 = 1125

## 5

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

a_{1}= 6; d= 2; n= 15.

a_{15} = 6 + 14 · 2 = 34

S_{15}= (6 + 34) · 15/2 = 300

## 6

Find the angles of a convex quadrilateral, knowing they are in arithmetic sequence and d = 25º.

The sum of the interior angles of a quadrilateral is 360º.

360= ( a_{1} + a_{4}) · 4/2

a_{4}= a_{1} + 3 · 25

360= ( a_{1} + a_{1} + 3 · 25) · 4/2

a_{1} = 105/2 = 52º 30' a_{2} = 77º 30'

a_{3} = 102º 30' a_{4} = 127º 30'

## 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; a_{3} = 8 + 2d

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

## 8

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

Central term x

1º x − d

3º x + d.

x − d + x + x + d = 27

x = 9

(9 − d)^{2} + 81 + (9 + d)^{2} = 511/2

d = ± 5 / 2

13/2, 9, 23/2

23/2, 9, 13/2