Question

a.

n	1	2	3	4	5
t(n)	150	120	90

this sequence is _________
es una secuencia
because the pattern is ________
porque el patrón es

b.

n	1	2	3	4	5
t(n)	80	40	20

this sequence is _________
es una secuencia
because the pattern is ________
porque el patrón es

Explanation:

Step1: Find sequence a's common difference

Calculate difference between terms: $120 - 150 = -30$, $90 - 120 = -30$

Step2: Fill sequence a's missing values

$t(4) = 90 + (-30) = 60$; $t(5) = 60 + (-30) = 30$

Step3: Identify sequence a's type

Linear (arithmetic) with $t(n)=180-30n$

Step4: Find sequence b's common ratio

Calculate ratio between terms: $\frac{40}{80} = \frac{1}{2}$, $\frac{20}{40} = \frac{1}{2}$

Step5: Fill sequence b's missing values

$t(4) = 20 \times \frac{1}{2} = 10$; $t(5) = 10 \times \frac{1}{2} = 5$

Step6: Identify sequence b's type

Exponential (geometric) with $t(n)=160\times(\frac{1}{2})^n$

Answer:

Part a.

$n$	$t(n)$
2	120
3	90
4	60
5	30

This sequence is arithmetic (linear) because the pattern is subtracting 30 each time (constant difference of -30).

Part b.

$n$	$t(n)$
2	40
3	20
4	10
5	5

This sequence is geometric (exponential) because the pattern is multiplying by $\frac{1}{2}$ each time (constant ratio of $\frac{1}{2}$).