Geometric sequences

1003134706

Level:

Find the second term and the common ratio of the geometric sequence

{a_{n}}_{n = 1}^{\infty}

, if:

\begin{aligned} a_{2} - a_{1} & = 22, \\ a_{3} - a_{2} & = 66. \end{aligned}

a_{2} = 33

q = 3

a_{2} = 11

q = 3

a_{2} = 22

q = 3

a_{2} = 33

q = 2

a_{2} = 11

q = 2

1003134705

Level:

Find the first term of the geometric sequence

{a_{n}}_{n = 1}^{\infty}

, if:

\begin{aligned} a_{3} \cdot a_{4} & = - 25, \\ a_{4} - a_{3} & = 10. \end{aligned}

- 5

5

- 1

1

- 15

1003134704

Level:

The first term of a geometric sequence is

3

. Find the common ratio so that the sum of the first

3

terms is minimal.

- \frac{1}{2}

- 5

0

2

- 1

1003134703

Level:

The first term of a geometric sequence is different from zero and

s_{3} = - 3 s_{2}

. (

s_{n}

is the sum of the first n terms.) Find the common ratio.

- 2

2

\frac{1}{2}

0

- 3

1003134702

Level:

The sum of the first three terms of a geometric sequence is

\frac{7}{8}

and the common ratio is

\frac{1}{2}

. Find the sum of all the terms from the

3

rd to the

5

th of the sequence.

\frac{7}{32}

\frac{31}{32}

\frac{3}{32}

\frac{7}{8}

\frac{5}{8}

1003134701

Level:

The sum of the first and the second term of a geometric sequence is

1

, the sum of the third and the fourth term is

4

and the common ratio is negative. Find the first term.

- 1

2

- 2

3

\frac{1}{3}

1003084910

Level:

We were given a geometric sequence

\frac{1}{2}, \frac{1}{4}, \dots

. What is the formula for the

n

th element of the sequence?

a_{n} = \frac{1}{2^{n}}, n \in N

a_{n} = \frac{1}{2^{n + 1}}, n \in N

a_{n} = \frac{1}{2^{n - 1}}, n \in N

a_{n} = \frac{1}{2^{2 n}}, n \in N

1003107308

Level:

The first five terms of a geometrical sequence are:

- 2, 1, - \frac{1}{2}, \frac{1}{4}, - \frac{1}{8}

. Find the recursive formula of this sequence.

a_{1} = - 2; a_{n + 1} = a_{n} \cdot (- \frac{1}{2}), n \in N

a_{1} = - 2; a_{n + 1} = a_{n} \cdot (- \frac{1}{4}), n \in N

a_{1} = - 2; a_{n + 1} = a_{n} \cdot \frac{1}{2}, n \in N

a_{1} = - 2; a_{n + 1} = a_{n} \cdot {(- \frac{1}{2})}^{n}, n \in N

9000072810

Level:

The following numbers form a geometric sequence. Find

x

x, 2 \cdot 3, 3 \cdot 4

1 \cdot 3

1 \cdot 1

1 \cdot 2

1 \cdot 4

9000073004

Level:

Consider a geometric sequence

(a_{n})_{n = 1}^{\infty}

. Let

q

be the quotient and

s_{n}

be the sum of the first

n

terms. Given

a_{1} = 1

a_{3} = 4

and

a_{2} < 0

, find the sum of the first four terms of the sequence.

s_{4} = - 5

s_{4} = 15

s_{4} = 14

s_{4} = 8

1003134706

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