Sure, let's break down the solution step-by-step.
### Given Sequences:
1. [tex]\( P = [10, 16, 22, P_4] \)[/tex]
2. [tex]\( Q = [5, 8, 11, Q_4] \)[/tex]
3. [tex]\( S = [15, 24, 33, S_4] \)[/tex]
It is given that the sequence [tex]\( S \)[/tex] is the sum of sequences [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]. Therefore, the relationship between these sequences can be expressed as:
[tex]\[ S = P + Q \][/tex]
### To find the missing numbers in the 4th row, [tex]\(P_4\)[/tex], [tex]\(Q_4\)[/tex], and [tex]\(S_4\)[/tex]:
Step 1: Determine [tex]\(P_4\)[/tex]
Observe the pattern in sequence [tex]\(P\)[/tex]:
- From 10 to 16, there is an increase of 6.
- From 16 to 22, there is another increase of 6.
Thus, the sequence is increasing by 6 each step. Therefore, for [tex]\(P_4\)[/tex]:
[tex]\[ P_4 = 22 + 6 = 28 \][/tex]
Step 2: Determine [tex]\(Q_4\)[/tex]
Observe the pattern in sequence [tex]\(Q\)[/tex]:
- From 5 to 8, there is an increase of 3.
- From 8 to 11, there is another increase of 3.
Thus, the sequence is increasing by 3 each step. Therefore, for [tex]\(Q_4\)[/tex]:
[tex]\[ Q_4 = 11 + 3 = 14 \][/tex]
Step 3: Determine [tex]\(S_4\)[/tex]
Since [tex]\( S = P + Q \)[/tex], for [tex]\( S_4 \)[/tex]:
[tex]\[ S_4 = P_4 + Q_4 \][/tex]
[tex]\[ S_4 = 28 + 14 = 42 \][/tex]
### The 4th row completed:
[tex]\[ P_4 = 28, \ Q_4 = 14, \ S_4 = 42 \][/tex]
Finally, filling in the numbers:
[tex]\[
\begin{array}{lll}
P & Q & S \\
\hline
10 & 5 & 15 \\
16 & 8 & 24 \\
22 & 11 & 33 \\
28 & 14 & 42 \\
\end{array}
\][/tex]
### Answer to part (a)(i):
The formula connecting [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( S \)[/tex] is:
[tex]\[ S = P + Q \][/tex]
### Answer to part (a)(ii):
It seems that part (a)(ii) is incomplete or improperly represented. If it relates to a general formula based on [tex]\( n \)[/tex]:
- If [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] increase linearly, the formulas for the [tex]\( n \)[/tex]-th term could be:
[tex]\[ P_n = 10 + 6(n-1) \][/tex]
[tex]\[ Q_n = 5 + 3(n-1) \][/tex]
- Then, the sequence [tex]\( S \)[/tex] would be:
[tex]\[ S_n = P_n + Q_n = (10 + 6(n-1)) + (5 + 3(n-1)) = 15 + 9(n-1) = 9n + 6 \][/tex]
Therefore, [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( S \)[/tex] have the following general forms:
[tex]\[ P_n = 6n + 4 \][/tex]
[tex]\[ Q_n = 3n + 2 \][/tex]
[tex]\[ S_n = 9n + 6 \][/tex]
