Answer:
Question 1
[tex]\textsf{Recursive}:\quad\begin{cases}f(n)=f(n-1)+2\\f(1)=12\end{cases}[/tex]
Value of the 50th term: 110
Explanation: The sequence is arithmetic. Each term is 2 more than the previous term. The first term is 12.
Question 2
[tex]\textsf{Recursive}: \quad \begin{cases}f(n)=2f(n-1)\\f(1)=2\end{cases}[/tex]
Value of the 7th term: 128
Explanation: The sequence is geometric. Each term is double the previous term. The first term is 2.
Step-by-step explanation:
An Arithmetic Sequence has a common difference between each term, so the difference between each term is the same.
A Geometric Sequence has a common ratio (multiplier) between each term, so each term is multiplied by the same number.
An Explicit Formula for a sequence allows you to find the nth term of the sequence.
A Recursive Formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Question 2
Given explicit equation:
[tex]f(n)=2n+10[/tex]
This is an arithmetic sequence.
Recursive Formula for an arithmetic sequence
[tex]f(n)=f(n-1)+d \quad \textsf{for }n \geq 2[/tex]
where:
- f(n) is the nth term.
- f(n - 1) is the term immediately before the nth term.
- d is the common difference between terms.
From inspection of the given table, the common difference between terms is:
[tex]\implies d=14-12=2[/tex]
Therefore, the Recursive Rule is:
[tex]\begin{cases}f(n)=f(n-1)+2\\f(1)=12\end{cases}[/tex]
Please note: When giving a recursive equation, the first term must be defined.
To find the value of the 50th term, substitute n = 50 into the explicit equation:
[tex]\implies f(50)=2(50)+10 = 110[/tex]
Explanation:
The sequence is arithmetic. Each term is 2 more than the previous term. The first term is 12.
Question 3
Given explicit equation:
[tex]f(n)=1(2)^n[/tex]
This is a geometric sequence.
Recursive Formula for a geometric sequence
[tex]f(n)=f(n-1) \times r \quad \textsf{for }n \geq 2[/tex]
where:
- f(n) is the nth term.
- f(n - 1) is the term immediately before the nth term.
- r is the common ratio between terms.
We are told that the sequence is being doubled at each step and that the 6th term is 64. Therefore, to find the first term, divide by 2:
[tex]\implies f(6)=64[/tex]
[tex]\implies f(5)=f(6) \div 2 = 64 \div 2 = 32[/tex]
[tex]\implies f(4)=f(5) \div 2 = 32 \div 2 = 16[/tex]
[tex]\implies f(3)=f(4) \div 2 = 16 \div 2 = 8[/tex]
[tex]\implies f(2)=f(3) \div 2 = 8 \div 2 = 4[/tex]
[tex]\implies f(1)=f(2) \div 2 = 4 \div 2 = 2[/tex]
Therefore, the Recursive Rule is:
[tex]\begin{cases}f(n)=2f(n-1)\\f(1)=2\end{cases}[/tex]
Please note: When giving a recursive equation, the first term must be defined.
To find the value of the 7th term, substitute n = 7 into the explicit equation:
[tex]\implies f(7)=1(2)^7=128[/tex]
Explanation:
The sequence is geometric. Each term is double the previous term. The first term is 2.
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