Question

topic: comparing additive and multiplicative patterns
the sequences below exemplify either an additive (arithmetic) or a multiplicative (geometric) pattern. identify the type of sequence, fill in the missing values on the table and write an equation.
1.

term	1st	2nd	3rd	4th	5th	6th	7th	8th
value	2	4	8	16	32	64

a. type of sequence:
b. equation:
2.

term	1st	2nd	3rd	4th	5th	6th	7th	8th
value	66	50	34	18

a. type of sequence:
b. equation:
3.

term	1st	2nd	3rd	4th	5th	6th	7th	8th
value	-3	9	-27	81

a. type of sequence:
b. equation:
4.

term	1st	2nd	3rd	4th	5th	6th	7th	8th
value	160	80	40	20

a. type of sequence:
b. equation:
5.

term	1st	2nd	3rd	4th	5th	6th	7th	8th
value	-9	-2	5	12

a. type of sequence:
b. equation:

Explanation:

1. First sequence (2, 4, 8, 16, 32, ...)

Step1: Identify type of sequence

Check the ratio between consecutive terms. $\frac{4}{2}=2$, $\frac{8}{4}=2$, $\frac{16}{8}=2$, $\frac{32}{16}=2$. It's a geometric sequence.

Step2: Find the 6th - 8th terms

The common ratio $r = 2$. The $n$th - term formula for a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1 = 2$. The 6th term is $a_6=2\times2^{6 - 1}=2^6 = 64$, the 7th term is $a_7=2\times2^{7 - 1}=2^7 = 128$, the 8th term is $a_8=2\times2^{8 - 1}=2^8 = 256$. The equation is $a_n=2\times2^{n - 1}=2^n$.

2. Second sequence (66, 50, 34, 18, ...)

Step1: Identify type of sequence

Find the common - difference. $50−66=-16$, $34 - 50=-16$, $18 - 34=-16$. It's an arithmetic sequence.

Step2: Find the 5th - 8th terms

The common difference $d=-16$. The $n$th - term formula for an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $a_1 = 66$ and $d=-16$. The 5th term is $a_5=66+(5 - 1)\times(-16)=66-64 = 2$, the 6th term is $a_6=66+(6 - 1)\times(-16)=66 - 80=-14$, the 7th term is $a_7=66+(7 - 1)\times(-16)=66-96=-30$, the 8th term is $a_8=66+(8 - 1)\times(-16)=66 - 112=-46$. The equation is $a_n=66+(n - 1)(-16)=66-16n + 16=82-16n$.

3. Third sequence (-3, 9, -27, 81, ...)

Step1: Identify type of sequence

Check the ratio between consecutive terms. $\frac{9}{-3}=-3$, $\frac{-27}{9}=-3$, $\frac{81}{-27}=-3$. It's a geometric sequence.

Step2: Find the 5th - 8th terms

The common ratio $r=-3$. The $n$th - term formula for a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1=-3$. The 5th term is $a_5=-3\times(-3)^{5 - 1}=-3\times81=-243$, the 6th term is $a_6=-3\times(-3)^{6 - 1}=-3\times(-243)=729$, the 7th term is $a_7=-3\times(-3)^{7 - 1}=-3\times729=-2187$, the 8th term is $a_8=-3\times(-3)^{8 - 1}=-3\times(-2187)=6561$. The equation is $a_n=-3\times(-3)^{n - 1}=(-3)^n$.

4. Fourth sequence (160, 80, 40, 20, ...)

Step1: Identify type of sequence

Check the ratio between consecutive terms. $\frac{80}{160}=\frac{1}{2}$, $\frac{40}{80}=\frac{1}{2}$, $\frac{20}{40}=\frac{1}{2}$. It's a geometric sequence.

Step2: Find the 5th - 8th terms

The common ratio $r=\frac{1}{2}$. The $n$th - term formula for a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1 = 160$. The 5th term is $a_5=160\times(\frac{1}{2})^{5 - 1}=160\times\frac{1}{16}=10$, the 6th term is $a_6=160\times(\frac{1}{2})^{6 - 1}=160\times\frac{1}{32}=5$, the 7th term is $a_7=160\times(\frac{1}{2})^{7 - 1}=160\times\frac{1}{64}=\frac{5}{2}$, the 8th term is $a_8=160\times(\frac{1}{2})^{8 - 1}=160\times\frac{1}{128}=\frac{5}{4}$. The equation is $a_n=160\times(\frac{1}{2})^{n - 1}$.

5. Fifth sequence (-9, -2, 5, 12, ...)

Step1: Identify type of sequence

Find the common - difference. $-2-(-9)=7$, $5-(-2)=7$, $12 - 5=7$. It's an arithmetic sequence.

Step2: Find the 5th - 8th terms

The common difference $d = 7$. The $n$th - term formula for an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $a_1=-9$ and $d = 7$. The 5th term is $a_5=-9+(5 - 1)\times7=-9 + 28 = 19$, the 6th term is $a_6=-9+(6 - 1)\times7=-9+35 = 26$, the 7th term is $a_7=-9+(7 - 1)\times7=-9 + 42 = 33$, the 8th term is $a_8=-9+(8 - 1)\times7=-9+49 = 40$. The equation is $a_n=-9+(n - 1)\times7=-9+7n - 7=7n-16$.

Answer:

1.
a. Geometric sequence
b. $a_n=2^n$
5th term: 32, 6th term: 64, 7th term: 128, 8th term: 256
2.
a. Arithmetic sequence
b. $a_n=82 - 16n$
5th term: 2, 6th term: -14, 7th term: -30, 8th term: -46
3.
a. Geometric sequence
b. $a_n=(-3)^n$
5th term: -243, 6th term: 729, 7th term: -2187, 8th term: 6561
4.
a. Geometric sequence
b. $a_n=160\times(\frac{1}{2})^{n - 1}$
5th term: 10, 6th term: 5, 7th term: $\frac{5}{2}$, 8th term: $\frac{5}{4}$
5.
a. Arithmetic sequence
b. $a_n=7n-16$
5th term: 19, 6th term: 26, 7th term: 33, 8th term: 40