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(a)
To prove: The given binomial theorem for n=1.
(b)
The statement for which one can assumed have statement is true and the statement which has to be proven. If the binomial expression is
(a+b)n=(n0)an+(n1)an−1b+(n2)an−2b2+⋯+(nn−1)abn−1+(nn)bn
Replace n=k and n=k+1.
(c)
To calculate: The statement for n=k that is assumed true to which (a+b) is multiplied and simplify.
(d)
To calculate: Collecting like terms on the right-hand side of the last statement, at what time one has
(a+b)k+1=(k0)ak+1+(k0)akb+(k1)akb+(k1)ak−1b2+(k2)ak−1b2+(k2)ak−2b3+⋯+(kk−1)a2bk−1+(kk−1)abk+(kk)abk+(kk)bk+1
(e)
To calculate: The addition of binomial sums in brackets as per the results of Exercise 84.
It is provided that
(a+b)k+1=(k0)ak+1+[(k0)+(k1)]akb+[(k1)+(k2)]ak−1b2+[(k2)+(k3)]ak−2b3+⋯+[(kk−1)+(kk)]abk+(kk)bk+1
(f)
The resultant statement that must be proved. The statement can be obtained by substituting the results of (k0)=(k+10)(why?) and (kk)=(k+1k+1) and the results of part e in to the equation of part d.
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Chapter 10 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Precalculus (6th Edition)
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