Geometric sequences

1003170604

Level:

Find the sum of the first ten natural powers of the number

2

2 046

2 048

2 047

2 000

1 536

1003170605

Level:

Suppose we insert number

10 530

between

2

unknown numbers such that they form

3

consecutive terms of a geometric sequence and their sum is

31 707

. What is the smaller of the two unknown numbers?

9 477

10 500

9 832

10 034

5 265

1003170606

Level:

Which of the following numbers cannot be a term of the geometric sequence given by its second term

a_{2} = 360

and the common ratio

q = \frac{3}{5}

250

216

600

\frac{1 944}{25}

\frac{648}{5}

1103170607

Level:

Let there be an equilateral triangle with a side length of

16 cm

. Connecting the centres of its sides with line segments we form another equilateral triangle. If we add three more triangles using the same procedure, what is the sum of their perimeters?

93 cm

72 cm

144 cm

31 cm

90 cm

1103170608

Level:

Let there be an equilateral triangle with a side length of

16 cm

. Connecting the centres of its sides with line segments we form another equilateral triangle. If we add two more triangles using the same procedure, what is the sum of their areas?

85 \sqrt{3} {cm}^{2}

128 \sqrt{3} {cm}^{2}

\frac{341}{4} \sqrt{3} {cm}^{2}

90 {cm}^{2}

148 \sqrt{3} {cm}^{2}

2010005507

Level:

A motorbike depreciates (loses value) at the rate of

12 %

per year. After how many full years will the value of the motorbike decrease to less than one third of its initial value?

9

6

7

8

10

2010005509

Level:

How many numbers do we need to insert between the numbers

5

and

640

so that the inserted numbers with the given two numbers are consecutive terms of a geometric sequence? The sum of all numbers inserted must be

630

6

4

3

5

7

2110014005

Level:

Between the roots of the quadratic equation

4 x^{2} - 35 x + 54 = 0

insert two numbers, so that the numbers with the roots together form four consecutive terms of a geometric sequence. This part of the geometric sequence is shown in one of the graphs. Choose the graph.