Math / Discrete

Question1)
Determine the first six terms of the sequence defined by $t_{1}=-5$ and $t_{n}=-3 x_{m-1}+8$ .

Studdy Solution

STEP 1

What is this asking? We need to find the first six numbers in a sequence where the first number is -5, and each following number is calculated using the previous one. Watch out! Don't forget that each term depends on the previous one, so we need to calculate them in order!

STEP 2

1. Find the second term
2. Find the third term
3. Find the fourth term
4. Find the fifth term
5. Find the sixth term
6. Review the terms

STEP 3

We're given that the first term is $t_1 = -5$ .
Awesome! The formula to find the next term in the sequence is $t_n = -3 \cdot t_{n-1} + 8$ .
So, to find the second term ( $t_2$ ), we'll plug in $n = 2$ into our formula.

STEP 4

That gives us $t_2 = -3 \cdot t_{2-1} + 8$ , which simplifies to $t_2 = -3 \cdot t_1 + 8$ .
Since $t_1 = -5$ , we substitute that in: $t_2 = -3 \cdot (-5) + 8$ .

STEP 5

Now, $-3 \cdot (-5)$ is 15 (because a negative times a negative is a positive).
So, $t_2 = 15 + 8 = \textbf{23}$ .
The second term is 23!

STEP 6

Now, let's find $t_3$ !
Using our formula with $n = 3$ , we have $t_3 = -3 \cdot t_{3-1} + 8$ , which is $t_3 = -3 \cdot t_2 + 8$ .
Remember, we just found that $t_2 = \textbf{23}$ , so we plug that in: $t_3 = -3 \cdot \textbf{23} + 8$ .

STEP 7

$-3 \cdot 23$ gives us $-69$ .
Then, $-69 + 8 = \textbf{-61}$ .
So, $t_3 = \textbf{-61}$ .

STEP 8

On to $t_4$ !
We have $t_4 = -3 \cdot t_3 + 8$ .
Substituting $t_3 = \textbf{-61}$ , we get $t_4 = -3 \cdot (\textbf{-61}) + 8$ .

STEP 9

$-3 \cdot (-61)$ is 183.
Adding 8, we get $183 + 8 = \textbf{191}$ .
Thus, $t_4 = \textbf{191}$ .

STEP 10

For $t_5$ , we use $t_5 = -3 \cdot t_4 + 8$ .
Plugging in $t_4 = \textbf{191}$ , we have $t_5 = -3 \cdot \textbf{191} + 8$ .

STEP 11

$-3 \cdot 191$ is $-573$ .
Then, $-573 + 8 = \textbf{-565}$ .
So, $t_5 = \textbf{-565}$ .

STEP 12

Finally, $t_6 = -3 \cdot t_5 + 8$ .
With $t_5 = \textbf{-565}$ , we have $t_6 = -3 \cdot (\textbf{-565}) + 8$ .

STEP 13

$-3 \cdot (-565)$ gives us 1695.
Adding 8, we get $1695 + 8 = \textbf{1703}$ .
Therefore, $t_6 = \textbf{1703}$ .

STEP 14

Let's recap!
We found: $t_1 = -5$ , $t_2 = 23$ , $t_3 = -61$ , $t_4 = 191$ , $t_5 = -565$ , and $t_6 = 1703$ .

STEP 15

The first six terms of the sequence are $-5$ , $23$ , $-61$ , $191$ , $-565$ , and $1703$ .

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