Question

learning goal: i can construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two input - output pairs. lesson reflection (circle one): starting..., getting there..., got it. complete the previous problems, check your solutions, then complete the lesson checkpoint below. complete the lesson reflection above by circling your current understanding of the learning goal(s). find the 5th term of the given geometric sequences. 1. 1, 8, 64, ... 2. 2000, 1000, 500, ... round your answer to the nearest hundredth of a unit if necessary. 3. the table shows the balance in an investment account after each month (t). the balances form a geometric sequence. what is the amount in the account at month 6? month amount 1 1700 2 2040 3 2448 4. a ball is dropped from the top of a building. the table shows its height in feet above ground at the top of each bounce. what is the height of the ball at the top of bounce 6? bounce height 1 250 2 200 3 160 4 128 5 102.4

Explanation:

Step1: Recall geometric - sequence formula

The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1$ is the first - term and $r$ is the common ratio.

Problem 1:

Step1: Find the common ratio $r$

Given the sequence $1,8,64,\cdots$, $a_1 = 1$ and $r=\frac{8}{1}=8$.

Step2: Calculate the 5th term

Using the formula $a_n=a_1r^{n - 1}$, for $n = 5$, $a_5=1\times8^{5 - 1}=8^4=4096$.

Problem 2:

Step1: Find the common ratio $r$

Given the sequence $2000,1000,500,\cdots$, $a_1 = 2000$ and $r=\frac{1000}{2000}=0.5$.

Step2: Calculate the 5th term

Using the formula $a_n=a_1r^{n - 1}$, for $n = 5$, $a_5=2000\times(0.5)^{5 - 1}=2000\times0.5^4=2000\times\frac{1}{16}=125$.

Problem 3:

Step1: Find the common ratio $r$

Given $a_1 = 1700$ and $a_2 = 2040$, $r=\frac{a_2}{a_1}=\frac{2040}{1700}=1.2$.

Step2: Calculate the 6th term

Using the formula $a_n=a_1r^{n - 1}$, for $n = 6$, $a_6=1700\times(1.2)^{6 - 1}=1700\times1.2^5=1700\times2.48832 = 4230.144\approx4230.14$.

Problem 4:

Step1: Find the common ratio $r$

Given $a_1 = 250$ and $a_2 = 200$, $r=\frac{a_2}{a_1}=\frac{200}{250}=0.8$.

Step2: Calculate the 6th term

Using the formula $a_n=a_1r^{n - 1}$, for $n = 6$, $a_6=250\times(0.8)^{6 - 1}=250\times0.8^5=250\times0.32768 = 81.92$.

Answer:

1. $4096$
2. $125$
3. $4230.14$
4. $81.92$