2. Define sequences (an) and (bn) by letting an-1+ bn-1 an = bn = Van-1bn-1- Deduce from the previous part that an > bn for all n, and hence also that an > an+1 and bn+1 > bn for all n.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
I need help with 2.
In #1 I already established that a1>b1
I just don’t know ho to elaborate for 2
![Let \( a_0 \) and \( b_0 \) be two positive real numbers with \( a_0 > b_0 \). The arithmetic mean of \( a_0 \) and \( b_0 \) is
\[
a_1 = \frac{a_0 + b_0}{2},
\]
and the geometric mean of \( a_0 \) and \( b_0 \) is
\[
b_1 = \sqrt{a_0 b_0}.
\]
1. For any values of \( a_0 \) and \( b_0 \), show that \( a_1 > b_1 \). This is a special case of the arithmetic mean - geometric mean (AMGM) inequality which says that the same inequality holds for the arithmetic and geometric mean of more than two numbers (you don’t need to prove this more general fact). Hint: start with the fact that \( (a_0 - b_0)^2 > 0 \).
2. Define sequences \( (a_n) \) and \( (b_n) \) by letting
\[
a_n = \frac{a_{n-1} + b_{n-1}}{2}, \quad b_n = \sqrt{a_{n-1}b_{n-1}}.
\]
Deduce from the previous part that
\[
a_n > b_n
\]
for all \( n \), and hence also that
\[
a_n > a_{n+1} \text{ and } b_{n+1} > b_n
\]
for all \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a238e39-c297-4642-8cf7-f442113dee2f%2Fa78e000c-9165-45bd-bf49-a0e919b3f1a2%2Fha674va_processed.jpeg&w=3840&q=75)
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