Sure! Let's solve each part of this question step-by-step.
### 6.1 Write down the values of the next 2 terms.
Given the sequence: 3, 9, 17, 27
First, we need to identify the pattern and determine the general term of the sequence. Once we have the general term, we can plug in the numbers to find the next terms.
### 6.2 Determine the general term of the sequence.
We suspect the sequence could follow a quadratic pattern. A general quadratic sequence can be written in the form:
[tex]\[ T(n) = an^2 + bn + c \][/tex]
To determine the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we use the first few terms of the sequence:
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[ T(1) = a(1)^2 + b(1) + c = a + b + c = 3 \][/tex]
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[ T(2) = a(2)^2 + b(2) + c = 4a + 2b + c = 9 \][/tex]
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[ T(3) = a(3)^2 + b(3) + c = 9a + 3b + c = 17 \][/tex]
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[ T(4) = a(4)^2 + b(4) + c = 16a + 4b + c = 27 \][/tex]
We now have a system of equations:
[tex]\[
\begin{cases}
a + b + c = 3 \\
4a + 2b + c = 9 \\
9a + 3b + c = 17 \\
\end{cases}
\][/tex]
Solving these equations:
After solving this system of equations, the values are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 3 \][/tex]
[tex]\[ c = -1 \][/tex]
Thus, our general term for the sequence is:
[tex]\[ T(n) = n^2 + 3n - 1 \][/tex]
### Finding the next two terms in the sequence:
Using the general term [tex]\( T(n) = n^2 + 3n - 1 \)[/tex] to find the 5th term ([tex]\( n = 5 \)[/tex]) and the 6th term ([tex]\( n = 6 \)[/tex]):
1. For [tex]\( n = 5 \)[/tex]:
[tex]\[ T(5) = 5^2 + 3(5) - 1 = 25 + 15 - 1 = 39 \][/tex]
2. For [tex]\( n = 6 \)[/tex]:
[tex]\[ T(6) = 6^2 + 3(6) - 1 = 36 + 18 - 1 = 53 \][/tex]
### Summary:
6.1 The next two terms are 39 and 53.
6.2 The general term of the sequence is:
[tex]\[ T(n) = n^2 + 3n - 1 \][/tex]
