1. Welcome To
A Session
Binomial Theorem

2. Business Mathematics Eleventh Edition BY
D.C. Sancheti  V.K. Kapoor
Chapter 10
Binomial Theorem
PowerPoint Presentation by S. B. Bhattacharjee

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Statement :
If n is a positive integer, then
Proof: (By the method of induction)
By actual multiplication we have
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Thus, the theorem is true for n = 2 and n = 3
We assume that the theorem is true for n = m ,
Multiplying both sides by (a+x), we have
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The expansions (1) and (2) are exactly of the same form.
Thus, if the theorem is true for n = m, it is true for n = m+1.
But we have seen that the theorem is true for n = 3
it is true for the value ( 3+1) i.e. 4 and so on.
Hence, the theorem is true for all positive integral values of n.
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General term in the expansion of (a+x)n
The general term in the expansion of (a+x)n is given by___
General term in the expansion of (1+x)n
The general term in the expansion of (1+x)n is given by ___
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Example 1
If the co-efficient of x2 and x3 in the expansion of
are equal
the value of k.
Let x2 occur in Tr+1 i.e. (r+1)th term
Since x2 occurs in Tr+1
xr =x2
r = 2
Again, let x3 occur in Tr+1
Tr+1 = 9cr.39-r. (kx)r = 9cr. 39-r.kr.xr
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Since x3 occurs in Tr+1
xr = x3
r = 3
Co-efficient of x3 = 9c k3 = 9c3.36.k3
According to the question,
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Example 2
If the 21st and 22nd terms in the expansion of (1+x)44 are equal, find the value of x.
21st term i.e. T21= T20+1 = 44c20.x20
22ndterm i.e. T22 = T21+1= 44c21.x21.
According to the question,
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Example 3
Find the term independent of x in the expansion of
Answer:
Let (r+1)th term i.e. Tr+1 be the required term.
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Since the term is independent of x, there must be no x.
(8+1)th i.e. 9th term is independent of x.
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Example 4
Find the co-efficient of
in the expansion of
Answer:
Let (r+1) term i.e. Tr+1 contain
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Since this term contains
Continued…
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Example - 5 a) b)
Continued…
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Continued…
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