a.
To prove:The nth pyramid number
for every natural number n .
a.
Explanation of Solution
Given information:
The first few pyramid number are
Formula used:Use the induction hypothesis such that if a statement is true for
Proof:
Since,
Consider the result is true for
Now assume the base is
Therefore, it is a pyramid with base
for every natural number n .
From the given statement,
Then,
This statement is true for
Consider the statement (1) is true for some natural number
Then,
Now prove that statement (2) is true
take
And
Therefore, statement (2) is true for
Hence, by induction hypothesis the statement is true for all-natural number n .
b.
To prove:The nth pyramid number
b.
Explanation of Solution
Given information:
The first few pyramid number are
Proof:
Consider for
Therefore,
c.
To prove:The nth pyramid number
for every natural number n .
c.
Explanation of Solution
Given information:
The first few pyramid number are
Proof:
Consider for every natural number n ,
Therefore,
d.
To prove:The nth pyramid number
d.
Explanation of Solution
Given information:
The first few pyramid number are
Proof:
Consider
Therefore,
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Chapter 2 Solutions
A Transition to Advanced Mathematics
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- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,