e sequence {an} is defined recursively as follows: = 2an-1+n + 5 for n > 1 = 4 e theorem below is proven by induction: eorem: for any non-negative integer n, an 11 27 — п — 7 hat must be established in the inductive step? For k >0 if ak = 11 · 2k – k – 7 then ak+1 2аk + (k + 1) + 5 For k >0 if ak 2ak-1 + k + 5 11· 2k+1 (k + 1) – 7 then ak+1 For k >0 if ak 2ak-1 + k + 5 then ak+1 2ak + (k + 1) + 5 For k >0 if ak 11 · 2* – k – 7 then ak+1 11 · 2k+1 – (k + 1) – 7
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
discrete math
![### Induction and Recursive Sequences: An Educational Overview
The sequence \(\{a_n\}\) is defined recursively as follows:
\[ a_n = 2a_{n-1} + n + 5 \quad \text{for} \quad n \geq 1 \]
\[ a_0 = 4 \]
We aim to prove the theorem below by induction:
### Theorem
For any non-negative integer \(n\),
\[ a_n = 11 \cdot 2^n - n - 7 \]
### Inductive Step
To employ mathematical induction, we assume the theorem holds for some arbitrary non-negative integer \(k\). That is, we assume:
\[ a_k = 11 \cdot 2^k - k - 7 \]
We must show the theorem holds for \(k+1\). This involves proving that:
\[ a_{k+1} = 11 \cdot 2^{k+1} - (k + 1) - 7 \]
Refer to the multiple-choice options to determine the correct steps to establish in the inductive step:
#### Options (Explanation via Reasoning):
1. **Option 1**:
\[ \text{For } k \geq 0 \text{, if } a_k = 11 \cdot 2^k - k - 7 \text{ then } a_{k+1} = 2a_k + (k + 1) + 5 \]
2. **Option 2**:
\[ \text{For } k \geq 0 \text{, if } a_k = 2a_{k-1} + k + 5 \text{ then } a_{k+1} = 11 \cdot 2^{k+1} - (k + 1) - 7 \]
3. **Option 3**:
\[ \text{For } k \geq 0 \text{, if } a_k = 2a_{k-1} - k + 5 \text{ then } a_{k+1} = 2a_k + (k + 1) + 5 \]
4. **Option 4**:
\[ \text{For } k \geq 0 \text{, if } a_k = 11](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5f558a7-14fc-4024-84d6-4debb1adc6f6%2Fe9909b04-ee8c-4acb-9d7c-45b571915c87%2F0u422nf_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
