Let's solve the given problem step-by-step. We need to find the first three terms of the sequence defined by [tex]\( a_n = -(10)^n \)[/tex] and then calculate the differences between these terms.
### Step 1: Define the Sequence
The general term of the sequence is given by:
[tex]\[ a_n = -(10)^n \][/tex]
### Step 2: Calculate the First Three Terms of the Sequence
#### First Term ([tex]\( n = 1 \)[/tex])
Plug in [tex]\( n = 1 \)[/tex] into the general term:
[tex]\[ a_1 = -(10)^1 = -10 \][/tex]
#### Second Term ([tex]\( n = 2 \)[/tex])
Plug in [tex]\( n = 2 \)[/tex] into the general term:
[tex]\[ a_2 = -(10)^2 = -100 \][/tex]
#### Third Term ([tex]\( n = 3 \)[/tex])
Plug in [tex]\( n = 3 \)[/tex] into the general term:
[tex]\[ a_3 = -(10)^3 = -1000 \][/tex]
So, the first three terms of the sequence are:
[tex]\[ -10, -100, -1000 \][/tex]
### Step 3: Calculate the Differences Between the Terms
#### Difference Between the First and Second Term
The difference between [tex]\( a_2 \)[/tex] and [tex]\( a_1 \)[/tex] is:
[tex]\[ a_2 - a_1 = -100 - (-10) = -100 + 10 = -90 \][/tex]
#### Difference Between the Second and Third Term
The difference between [tex]\( a_3 \)[/tex] and [tex]\( a_2 \)[/tex] is:
[tex]\[ a_3 - a_2 = -1000 - (-100) = -1000 + 100 = -900 \][/tex]
### Summary of Results
The first three terms of the sequence are:
[tex]\[ -10, -100, -1000 \][/tex]
The differences between these terms are:
[tex]\[ -90 \][/tex] (difference between the first and second term)
[tex]\[ -900 \][/tex] (difference between the second and third term)
Therefore, the differences between the first three terms are:
[tex]\[ (-90, -900) \][/tex]
These are the calculated values for the first three terms and their differences.
