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

Question

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

Accepted Answer

∻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).⚹◊What is the explicit formula for the nth term of a sequence that begins with 3, 5, 7?⍭ Generate me a similar question◊

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