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# Geometric Progression

1.Characterize a geometric progression:

Solution:

A progression (a_{n})^{∞}_{n=1} is told to be geometric if and only if exists such q є R real number; q ≠ 1, that for each n є N stands a_{n+1} = a_{n}.q. Number q is called a geometric progression ratio.

Properties:

a) a_{n} = a_{1}.q^{n-1}

b) a_{r} = a_{s}.q^{r-s}

c)

d) Stable incrementation:

e) Stable decrementation:

f) Sum of an infinite geometric progression: q < 1

2.Determine first 6 members of a geometric progression if stands a_{3} = 8 and a_{7} = 128.

Solution:

3.If a number is added to 2, 16 and 58, it results in first 3 geometric progression members. Find out the number and enumerate first 6 members of the progression.

4.Insert 4 numbers between the roots of the equation x^{2} -66x +128 = 0 so that they would make a geometric progression.

5.Enumerate first 6 progression members of a geometric progression that fits following conditions:

6.The surface area of a cuboid equals S = 78 cm^{2}. The sides of the cuboid make a geometric progression. Sum of the lengths of the sides intersecting in one of the edges is 13 cm. Determine the volume of the cuboid.

The sides lengths are a = 1cm, b = 3cm, c = 9cm.

The volume equals V = a.b.c

V = 1.3.9 cm

^{3}

V = 27 cm

^{3}

7.An equestrian wants to buy a horse for $10,000. He made a deal with the salesman to pay for the nails in the horseshoes. He paid 1 cent for the first nail, 2 cents for the second nail, 4 cents for the third nail and so on. Each horseshoe is fastened by 5 nails. Did he make a good deal?

The equestrian overbid the horse for $485,75.

8.A workman agreed to work under following conditions: His salary for the first day of work will be $1, for the second day of work $2, for the third day of work $4, and so on. How long does he have to work to earn $4095?

The workman needs to work for 12 days.

9.What should be the bank interest to raise the deposit of $10000 to $25 000 in 5 years?

The interest should be 20%.

10.What is the sum of following infinite geometric progression?

The sum of the progression members is 3/2.

11. Solve in real numbers:

12. Solve in real numbers: