Chapter 44, Binomial Theorem Video Solutions, Schaum's outlines: precalculus

Problem 1

Calculate the binomial coefficients:
(a) $\left(\begin{array}{l}4 \\ 2\end{array}\right)$;
; (b) $\left(\begin{array}{l}8 \\ 5\end{array}\right)$;
; (c) $\left(\begin{array}{c}12 \\ 1\end{array}\right)$;
;(d) $\left(\begin{array}{c}n \\ n-1\end{array}\right)$

Ruby P

Numerade Educator

01:42

Problem 2

Show that $\left(\begin{array}{l}n \\ n\end{array}\right)=\left(\begin{array}{l}n \\ 0\end{array}\right)=1$.

Anna Jones

Numerade Educator

04:42

Problem 3

Show that $\left(\begin{array}{l}n \\ r\end{array}\right)=\frac{n(n-1) \cdot \cdots \cdot(r+1)}{(n-r) !}=\frac{n(n-1) \cdots \cdots \cdot(n-r+1)}{r !}$ for any integer $r<n$.

Joanna Quigley

Numerade Educator

03:24

Problem 4

Use the results of the previous problems to write out the terms of $(a+b)^4$.

Shenade Gordon

Numerade Educator

03:24

Problem 5

Use the results of the previous problems to write out the terms of $(a+b)^4$.

Shenade Gordon

Numerade Educator

01:27

Problem 6

Write the first three terms in the binomial expansion of $(a+b)^{20}$.

Tony Ni

Numerade Educator

01:18

Problem 7

Write the first three terms in the binomial expansion of $\left(2 x^5+3 t^2\right)^{12}$.

Amy Jiang

Numerade Educator

00:54

Problem 8

Show that $\left(\begin{array}{l}n \\ r\end{array}\right)=\left(\begin{array}{c}n \\ n-r\end{array}\right)$.

Linh Vu

Numerade Educator

02:12

Problem 9

Show that $\left(\begin{array}{c}k \\ r-1\end{array}\right)+\left(\begin{array}{l}k \\ r\end{array}\right)=\left(\begin{array}{c}k+1 \\ r\end{array}\right)$.

Linh Vu

Numerade Educator

00:47

Problem 10

Show that the binomial coefficients can be arranged in the form shown in Fig. 44-1.
$\begin{gathered}1 \\ 11 \\ 121 \\ 1331 \\ 14641\end{gathered}$

Ankit Gupta

Numerade Educator

05:49

Problem 11

Use mathematical induction to prove the binomial theorem for positive integers $n$.

Prabhat Tyagi

Numerade Educator

01:26

Problem 12

Write the eighth term in the expansion of $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{13}$.

Ankit Gupta

Numerade Educator

01:28

Problem 13

Use the binomial theorem to approximate $(1.01)^{20}$ to three decimal places.

Ankit Gupta

Numerade Educator

01:16

Problem 14

Calculate the binomial coefficients: (a) $\left(\begin{array}{c}15 \\ 1\end{array}\right)$;
(b) $\left(\begin{array}{l}8 \\ 6\end{array}\right)$;
(c) $\left(\begin{array}{c}12 \\ 9\end{array}\right)$
; (d) $\left(\begin{array}{c}n \\ n-2\end{array}\right)$

Tyler Moulton

Numerade Educator

02:45

Problem 15

Write the binomial expansion of (a) $(a+b)^5$; (b) $(2 x+y)^5$.

Tony Ni

Numerade Educator

03:35

Problem 16

Write the binomial expansion of (a) $(4 s-3 t)^3$; (b) $\left(2 a-\frac{b}{5}\right)^5$.

Angela Guo

Numerade Educator

Problem 17

Prove that $\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\cdots+\left(\begin{array}{c}n \\ n-1\end{array}\right)+\left(\begin{array}{l}n \\ n\end{array}\right)=2^n$, that is, that the sum of the binomial coefficients for any power $n$ is equal to $2^n$.

Check back soon!

01:13

Problem 18

Find the middle term in the binomial expansion of (a) $\left(3 x-\frac{y}{3}\right)^{14}$; (b) $\left(x^3+2 y^3\right)^{10}$.

Joanna Quigley

Numerade Educator

Problem 19

It is shown in calculus that if $|x|<1$ and $\alpha$ is not a positive integer, then $(1+x)^\alpha=\sum_{j=0}^{\infty}\left(\begin{array}{l}\alpha \\ j\end{array}\right) x^j$ with $\left(\begin{array}{l}\alpha \\ j\end{array}\right)=\frac{\alpha(\alpha-1) \cdot \cdots \cdot(\alpha-j+1)}{j !}$. Use this formula to write the first three terms of the binomial expansion of (a) $(1+x)^{-2}$; (b) $(1+x)^{1 / 2}$.

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