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# Arithmetic Progression

1.Characterize an aritmetic progression:

A progression (a_{2})^{∞}_{n+1 }is told to be „arithmetic“ if and only if exists such d є R real number, so that for all n є N stands a_{n+1} = a_{n} + d

Number d is called arithmetic progression difference.

For an arithmetic progression stands:

- a
_{n}= a_{1}+ (n-1)d - a
_{r}= a_{s}+ (r-s)d - s = a
_{1}+ a_{2}+ a_{3}+ ... + a_{n}= - If d > 0 the progression is increasing
- If d < 0 the progression is decreasing
- If d = 0 the progression is constant

2.Find out whether the following progression is arithmetic:

a

_{1}= -2, a

_{2}= -1, a

_{3}= 0, a

_{4}= 1, a

_{5}= 2 etc.

The progression is arithmetic; difference d = 1.

3.Enumerate first 6 progression members of an arithmetic progression that fits following conditions:

_{1}+ a

_{4}+ a

_{6}= 71

a

_{5}– a

_{2}– a

_{3}= 2

a

_{1}+ a

_{1}+3d +a

_{1}+5d = 71

a

_{1}+ 4d – a

_{1}-d –a

_{1}-2d = 2

3a

_{1}+ 8d = 71

-a

_{1}+ d =2/3

3a

_{1}+ 8d = 71

-3a

_{1}+ 3d = 6

a

_{1}– d = -2

a

_{1}= d - 2

a

_{1}= 7 - 2

a

_{1}= 5

11d = 77

d = 7

(a

_{2})

_{n-1}

^{6}= 5;12;19;26;33;40

4.Enumerate first 6 progression members of an arithmetic progression that fits following conditions:

5.An aritmetic progression consists of 8 numbers. Sum of the middle members is 41, sum of the first and the last member is 114. Enumerate the progression members.

6.Insert 4 numbers between the roots of the quadratic equation x2 – 16x +39 = 0 so that they will make an arithmetic progression.

Numbers to insert: 5;7;9;11

7.The lengths of the sides of a right triangle make an arithmetic progression. The longer cathetus is 24 cm long. Find out the perimeter of the triangle.

The perimeter of the triangle equals 72 cm.

8.Find out the sum of integers 1+2+3+4+5+6+7+.............+100.

Sum of the numbers S = 5050.

9.Iron pipes are stored in 8 rows. The top row contains of 13 pipes, next row always contains of one more pipe than the previous one. How many pipes are there?

There are 132 pipes.

10.The angles in a triangle make an arithmetic progression. The smallest angle is 20°. Determine sizes of the other angles.

The angles are 20°, 60° and 100°.

11.Sizes of the sides of a cuboid make 3 members of an arithmetic progression. Sum of the sizes equals 24 cm, the volume of the cuboid equals 312 cm^{3}. Determine the sizes of the sides.

The sizes of the sides are 3 cm, 8 cm and 13cm.

12.How long would a rock be falling to a mine somewhere in South America which is deep 2500 m if it moves 4,904 m in the first second and extra 9,808 m for every next second?

The rock would fall for approximately 22,5 sec.