Algebra II - Geometric Sequences

Given the sequence , what is the 7th term?

Explanation

The formula for geometric sequences is defined by:

$ar^{n-1}$

The term represents the first term, while is the common ratio. The term represents the terms.

Substitute the known values.

$3(3)^{n-1}$

To determine the seventh term, simply substitute into the expression.

$3(3)^{7-1}$

The answer is:

$\left { -1, 2, -4, 8, -16, 32... \right }$

What is the explicit formula for the above sequence? What is the 20th value?

$a_{n}= -1\cdot -2^{n-1};\ a_{20} = 524288$

$a_{n}=a_{n-1}\cdot-1;\ a_{20}=524288$

$a_{n}=a_{n-1}\cdot-2^{n-1};\ a_{20} = -262144$

$a_{n-1}= a_{n}\cdot -2;\ a_{20}=-524288$

$a_{n}=a_{1}\cdot-2;\ a_{20}=-524288$

Explanation

This is a geometric series. The explicit formula for any geometric series is:

$a_{n}= a_{1} \cdot r^{n-1}$ , where is the common ratio and is the number of terms.

In this instance and .

$a_{n}= a_{1} \cdot r^{n-1}\rightarrow a_n=(-1)\cdot(-2)^{n-1}$

Substitute into the equation to find the 20th term:

$a_{20}=(-1)\cdot(-2)^{20-1}$

$a_{20}=-(-2)^{19}=-(-524288)=524288$

Which of the following is a geometric sequence?

$t_{0}=4$

$t_{n}=t_{n-1}\times 3$

$t_{0}=5$

$t_{n}=t_{n-1}+3$

$t_{0}=2$

$t_{n} = t_{n-1}^{6}$

$t_{0}=10$

$t_{n}=t_{n-1}+5$

$t_{0}=4$

$t_{n}=t_{n-1}+3.4$

Explanation

A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. Thus, we look for an implicit definition which involves multiplication of the previous term. The only possibility is:

$t_{0}=4$

$t_{n}=t_{n-1}\times 3$

What type of sequence is shown below?

$\left { 4,9,15,22,30... \right }$

None of the other answers

Arithmetic

Geometric

Subtractive

Multiplicative

Explanation

This series is neither geometric nor arithmetic.

A geometric sequences is multiplied by a common ratio () each term. An arithmetic series adds the same additional amount () to each term. This series does neither.

Mutiplicative and subtractive are not types of sequences.

Therefore, the answer is none of the other answers.

Find the 19th term of the sequence $7,000, \enspace 2,800, \enspace 1,120, \enspace 448, ...$

\[the first term is 7,000\]

Explanation

First find the common ratio by dividing the second term by the first:

$2,800 \div 7,000 = \frac{ 2}{5} = 0.4$

Since the first term is , the nth term can be found using the formula

$\a_{n} = a_1(r)^{n-1}$

$7,000(0.4)^ {n-1 }$ ,

so the 19th term is $7,000(0.4)^{18} = 0.000481$

Which of the following could be the formula for a geometric sequence?

$a_6=3\cdot 7^5, a_1=3$

$a_{16}=a_{15}\cdot 3$

$a_4=5\cdot (\frac{1}{3})^4$

Explanation

The explicit formula for a geometric series is $a_n=a_1\cdot r^{n-1}$ .

Therefore, ${\color{Red} a_6=3\cdot 7^5, a_1=3}$ is the only answer that works.

$\textup{In the above geometric sequence, what is the value of }n\textup{?}$

Explanation

$\textup{Look for a pattern. Each term in the sequence is three times the previous term.}$

$18\ast3=54$

Find the 26th term of the sequence $1,771,561, \enspace -1,449,459, \enspace 1,185,921, \enspace -970,299, ...$

Explanation

First we need to find the common ratio, which we can do by dividing the second term by the first:

$-1,449,459 \div 1,771,561 = 0.\overline{81}$

The first term is $1,771, 561 (-0.\overline{81} ) ^ 0$ , the second term is $1,771, 561 (-0.\overline{81})^ 1$ , so the 26th term is $1,771, 561 (-0.\overline{81})^{ 25} = -11,738.206$