I need assiatance to understand a problem in Inequality

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In summary, the proof shows that for any positive integer n, the inequality (1+1/n)^n < (1+1/n+1)^n+1 is true, using the fact that for any positive real numbers a and b, with a < b, we can show that (n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n for each k, where 0≤k≤n. This is derived from the repeated use of a < b.
  • #1
sabyakgp
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Hello Friends,

I at a loss to understand the parts of the following proof:
For any positive ineteger n, prove that:

(1+1/n)^n < (1+1/n+1)^n+1
a, b positive real numbers such that a < b

Proof:

b^n+1 - a^n+1 = (b-a)(b^n+ab^n-1+...+a^n)
I could not understand the following part:
By a repeated use of a < b
(n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n
.
.
.
How the inequality equation (n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n
is derived from a < b?
I can understand how (n+1)a^n < (n+1) b^n, but how (n+1)b^n is greater than (b^n+ab^n-1+...+a^n) and how it's greater than a^n?
Could anyone please help me?

Best Regards,
Sabya
 
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  • #2
an < akbn-k < bn for each k, where 0≤k≤n, since a < b. There are n+1 terms in the sum, so the final inequalities hold.
 

1. What is inequality?

Inequality refers to the unequal distribution of resources, opportunities, and privileges among individuals or groups based on factors such as race, gender, socioeconomic status, or other characteristics.

2. How does inequality manifest in society?

Inequality can manifest in various ways, including income disparity, access to education and healthcare, employment opportunities, housing segregation, and unequal treatment in the criminal justice system.

3. What are the consequences of inequality?

The consequences of inequality can be far-reaching and damaging, leading to social and economic instability, increased poverty and crime rates, and hindering overall societal progress and development.

4. What are some causes of inequality?

Inequality can stem from systemic and structural issues, such as historical and ongoing discrimination, unequal distribution of resources and opportunities, and policies that favor certain groups over others.

5. How can we address and reduce inequality?

Addressing and reducing inequality requires a multifaceted approach, including promoting policies that promote equal opportunities and access to resources, addressing systemic issues, and promoting education and awareness about the impacts of inequality.

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