```
1.Find simple rules or formulas that define at least two different sequences that begin with the terms 3,5,7. Can you give recursive and non-recursive definitions for these sequences?
2.Use a loop invariant to prove that the following program segment, for computing the nth power of a real number x is correct (n greater or equal than 1n greater or equal than 1):
power: = 1
i:= 1
while i less or equal than ni less or equal than n
power := power * x
i : = i + 1
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Identify the given sequence and determine the pattern. The sequence begins with 3, 5, 7

step 2

For the first sequence, we can define a simple rule where each term is 2 more than the previous term. This is an arithmetic sequence

step 3

The non-recursive formula for the first sequence is $a_n = 2n + 1$, where $n$ starts at 1

step 4

The recursive formula for the first sequence is $a_1 = 3$ and $a_{n+1} = a_n + 2$ for $n \geq 1$

step 5

For the second sequence, we can define a rule where each term is the previous term multiplied by a constant. Let's choose the constant to be 1.5 for this example

step 6

The non-recursive formula for the second sequence is $a_n = 3 \cdot (1.5)^{n-1}$

step 7

The recursive formula for the second sequence is $a_1 = 3$ and $a_{n+1} = 1.5 \cdot a_n$ for $n \geq 1$

Answer

First sequence non-recursive: $a_n = 2n + 1$, recursive: $a_1 = 3$, $a_{n+1} = a_n + 2$. Second sequence non-recursive: $a_n = 3 \cdot (1.5)^{n-1}$, recursive: $a_1 = 3$, $a_{n+1} = 1.5 \cdot a_n$.

Key Concept

Arithmetic and Geometric Sequences

Explanation

The first sequence is arithmetic with a common difference of 2. The second sequence is geometric with a common ratio of 1.5.

Solution by Steps

step 1

Understand the program segment and identify the loop invariant to be used for the proof

step 2

Define the loop invariant: At the start of each iteration of the while loop, $power = x^{i-1}$

step 3

Initialization: Before the loop starts, $i=1$ and $power=1$, which is consistent with the loop invariant since $power = x^{1-1} = x^0 = 1$

step 4

Maintenance: Assume the loop invariant holds at the start of an arbitrary iteration, i.e., $power = x^{i-1}$. After executing the body of the loop, $power$ is updated to $power \cdot x$, which is $x^{i-1} \cdot x = x^i$. Incrementing $i$ by 1 maintains the invariant

step 5

Termination: The loop terminates when $i > n$. At this point, the loop invariant gives us $power = x^{i-1} = x^n$, which is the correct final value

Answer

The program segment correctly computes $x^n$ for $n \geq 1$.

Key Concept

Loop Invariant

Explanation

The loop invariant helps prove the correctness of a loop. It must hold true before the loop starts (initialization), remain true before each iteration (maintenance), and upon loop termination, it should provide a useful property that helps to show the loop's correctness (termination).

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