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Challenges

(1.2) Geometric Sequences

Exploring Geometric Sequences

For the sequence

$2,4,8,16,32,...$ , what number is multiplied each time to get the next term?

Find a formula for the general term,

$u_n$ of

$2,4,8,16,32,...$
Write your answer in the form

$b^n$

$u_n=$

Find a formula for the general term,

$u_n$ of

$1,2,4,8,16,32,...$
Write your answer in the form

$b^{n-1}$

$u_n=$

Find a formula for the general term,

$u_n$ of

$3,6,12,24,48,96,...$
Write your answer in the form

$a\left(b\right)^{n-1}$

$u_n=$

Find a formula for the general term,

$u_n$ of

$6,12,24,48,96,...$
Write your answer in the form

$a\left(b\right)^{n-1}$

$u_n=$

Find a formula for the general term,

$u_n$ of

$-1,-2,-4,-8,-16,-32...$
Write your answer in the form

$b^{n-1}$

$u_n=$

Find a formula for the general term,

$u_n$ of

$1,-2,4,-8,16,-32...$
Write your answer in the form

$(b)^{n-1}$

$u_n=$

Find a formula for the general term,

$u_n$ of

$-3,6,-12,24,-48,96,...$
Write your answer in the form

$a\left(b\right)^{n-1}$

$u_n=$

Terms of Geometric Sequences

Find the first 5 terms for each geometric sequence.
Write the terms separated by commas.
For example,

Question:

$u_n=5^n$

Answer:

$5,25,125,625,3125$

$u_n=3^n$

$u_n=3^{n-1}$

$u_n=2\left(3\right)^{n-1}$

$u_n=-2\left(3\right)^{n-1}$

$u_n=2\left(-3\right)^{n-1}$

$u_n=-2\left(-3\right)^{n-1}$

The $n^{th}$ Term of a Geometric Sequence

For geometric sequences, the number that is multiplied each time to find the next term is called the . It is usually denoted by the letter .

$u_1\overset{\times r}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_2}{\text{___}}\overset{\times r}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_3}{\text{___}}\overset{\times r}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_4}{\text{___}}\overset{\times r}{\overset{\Huge\frown}{\phantom{A},\phantom{B}}}\underset{u_5}{\text{___}}\phantom{A},\quad\ldots$ Let's find the terms of a geometric sequence starting from

$u_1$ :

$\qquad u_2 = u_1 r$

$\qquad u_3 = u_1 r^2$

$\qquad u_4 = u_1 r^3$
Follow this pattern and express

$u_5$ and

$u_6$ in terms of

$u_1$ and

$r$

$\qquad u_5 =$

$\qquad u_6 =$

To find

$u_n$ , how many

$r$ 's must be multiplied to

$u_1$ ?

For geometric sequences,

$u_n=$ .

General Term of Geometric Sequences

Find the general term, $u_n$ , for the following sequences.
*Write your answers in the form $b^{n-1}$ or $\left(b\right)^{n-1}$ or $a\left(b\right)^{n-1}$ or $a\left(\frac{b}{c}\right)^{n-1}$ where $a,b,c$ are numbers.

$2,6,18,54,...\qquad u_n=$

$10,50,250,1250...\qquad u_n=$

$1,-3,9,-27,...\qquad u_n=$

$32,16,8,4,2,...\qquad u_n=$

$12,18,27,\frac{81}{2},...\qquad u_n=$

$\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},...\qquad u_n=$

$-64,16,-4,1,-\frac{1}{4},...\qquad u_n=$

A geometric sequence has $u_2=-6$ and $u_5=162$ . Find the general term in the form $a\left(b\right)^{n-1}$ where $a, b$ are integers.

Find the first term of the sequence $6,6\sqrt{2},12,12\sqrt{2},…$ which exceeds $1400$ .

$k-1, 2k$ , and $21-k$ are consecutive terms of a geometric sequence. Find the two values of $k$ . Separate the answers with a comma.

Back(1.2) Geometric Sequences

Exploring Geometric Sequences

Terms of Geometric Sequences

The nthn^{th} Term of a Geometric Sequence

General Term of Geometric Sequences

(1.2) Geometric Sequences

The $n^{th}$ Term of a Geometric Sequence