Question

5-87. write the first five terms of each sequence. show all work.
a. t(1) = -3
t(n + 1) = -2·t(n)
b. t(1) = 8
t(n + 1) = t(n) - 5
c. t(1) = 2
t(n + 1) = (t(n))⁻¹

Explanation:

Part a

Step1: Find \( t(1) \)

Given \( t(1) = -3 \)

Step2: Find \( t(2) \)

Using \( t(n + 1) = -2 \cdot t(n) \), for \( n = 1 \):
\( t(2) = -2 \cdot t(1) = -2 \cdot (-3) = 6 \)

Step3: Find \( t(3) \)

For \( n = 2 \):
\( t(3) = -2 \cdot t(2) = -2 \cdot 6 = -12 \)

Step4: Find \( t(4) \)

For \( n = 3 \):
\( t(4) = -2 \cdot t(3) = -2 \cdot (-12) = 24 \)

Step5: Find \( t(5) \)

For \( n = 4 \):
\( t(5) = -2 \cdot t(4) = -2 \cdot 24 = -48 \)

Step1: Find \( t(1) \)

Given \( t(1) = 8 \)

Step2: Find \( t(2) \)

Using \( t(n + 1) = t(n) - 5 \), for \( n = 1 \):
\( t(2) = t(1) - 5 = 8 - 5 = 3 \)

Step3: Find \( t(3) \)

For \( n = 2 \):
\( t(3) = t(2) - 5 = 3 - 5 = -2 \)

Step4: Find \( t(4) \)

For \( n = 3 \):
\( t(4) = t(3) - 5 = -2 - 5 = -7 \)

Step5: Find \( t(5) \)

For \( n = 4 \):
\( t(5) = t(4) - 5 = -7 - 5 = -12 \)

Step1: Find \( t(1) \)

Given \( t(1) = 2 \)

Step2: Find \( t(2) \)

Using \( t(n + 1) = (t(n))^{-1}=\frac{1}{t(n)} \), for \( n = 1 \):
\( t(2) = \frac{1}{t(1)}=\frac{1}{2} \)

Step3: Find \( t(3) \)

For \( n = 2 \):
\( t(3) = \frac{1}{t(2)}=\frac{1}{\frac{1}{2}} = 2 \)

Step4: Find \( t(4) \)

For \( n = 3 \):
\( t(4) = \frac{1}{t(3)}=\frac{1}{2} \)

Step5: Find \( t(5) \)

For \( n = 4 \):
\( t(5) = \frac{1}{t(4)} = 2 \)

Answer:

(First five terms for part a): \(-3, 6, -12, 24, -48\)

Part b