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Write the fifteenth term of the binomial expansion of (a^2+b)^20

Write the fifteenth term of the binomial expansion of (a^2+b)^20-example-1
User Akane
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Answer:

The fifteenth term of the binomial expansion of
(a+b)^(20) is
38760\cdot a^(6)\cdot b^(14).

Explanation:

Let be a binomial of the form
(a+b)^(n), where
a, b\in \mathbb{R} and
n\,\in\mathbb{N}^(+). The expansion of this polynomial is defined below:


(a+b)^(n) = \Sigma\limits_(k=0)^(n)\,(n!)/(k!\cdot (n-k)!)\cdot (a^(n-k)\cdot b^(k)) (1)

Where:


n - Number of terms of the expanded polynomial.


k - Index associated to
k-th term of the expanded polynomial.

For all
n-th binomial, we a sum of
n+1 terms. If the given binomial has a term of
20, then we have 21 terms and the fifteenth term of the polynomial corresponds to the
14-th term. Then, the fifteenth term of the binomial is:


c_(14) = (20!)/(14!\cdot 6!)\cdot (a^(6)\cdot b^(14))


c_(14) = 38760\cdot a^(6)\cdot b^(14)

The fifteenth term of the binomial expansion of
(a+b)^(20) is
38760\cdot a^(6)\cdot b^(14).

User TaylorAllred
by
8.1k points

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