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How do you calculate the coefficients of a power series such as this one: [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10. Just the first 3 coefficients, with steps, would be great. I would greatly appreciate any help. Thanks a million and have a great day.

16.05.2016

#1
+14537
0

Hello !

[ 1/4x^4 + 1/4x^5 + 1/4x^6]^10

What do you mean ?

[ 1/4x^4 + 1/4x^5 + 1/4x^6]^10  =  $$\frac{(x^4+x^5+x^6)^{10}}{1048576}$$

or    [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10 = 0         =>   x = 0      and   $$x=-\sqrt[3]{-1}$$

17.05.2016
#2
+22188
0

How do you calculate the coefficients of a power series such as this one:
[ 1/4x^4 + 1/4x^5 + 1/4x^6]^10. Just the first 3 coefficients, with steps, would be great.
I would greatly appreciate any help. Thanks a million and have a great day.

$$\begin{array}{rcll} && [~ \frac14 \cdot x^4 + \frac14 \cdot x^5 + \frac14 \cdot x^6 ~]^{10} \\ &=& [~ \frac{1}{4} \cdot ( x^4 + x^5 + x^6 ) ~]^{10} \\ &=& \left( \frac{1}{4} \right)^{10} \cdot ( x^4 + x^5 + x^6 )^{10} \\ &=& \left( \frac{1}{4^{10} } \right) \cdot ( x^4 + x^5 + x^6 )^{10} \\ &=& \left( \frac{1}{1048576} \right) \cdot ( x^4 + x^5 + x^6 )^{10} \\\\ && ( a + b + c )^{n} \\ &=& \sum \limits_{k=0}^{n} \sum \limits_{i=0}^{k} \dbinom{n}{k} \dbinom{k}{i} a^{n-k} b^{k-i} c^{i} \qquad | \qquad (n-k)+(k-i)+(i) = n \\\\ && ( x^4 + x^5 + x^6 )^{10} \\ &=& \sum \limits_{k=0}^{10} \sum \limits_{i=0}^{k} \dbinom{10}{k} \dbinom{k}{i} (x^4)^{10-k} (x^5)^{k-i} (x^6)^{i} \qquad | \qquad (10-k)+(k-i)+(i) = 10 \\\\ \end{array}$$

$$\begin{array}{lcll} (k=0,i=0): & \dbinom{10}{0} \dbinom{0}{0} (x^4)^{10} (x^5)^{0} (x^6)^{0} &=& x^{40} \\ \hline (k=1,i=0): & \dbinom{10}{1} \dbinom{1}{0} (x^4)^{9} (x^5)^{1} (x^6)^{0} &=& 10 \cdot x^{41} \\ (k=1,i=1): & \dbinom{10}{1} \dbinom{1}{1} (x^4)^{9} (x^5)^{0} (x^6)^{1} &=& 10 \cdot x^{42} \\ \hline (k=2,i=0): & \dbinom{10}{2} \dbinom{2}{0} (x^4)^{8} (x^5)^{2} (x^6)^{0} &=& 45 \cdot x^{42} \\ (k=2,i=1): & \dbinom{10}{2} \dbinom{2}{1} (x^4)^{8} (x^5)^{1} (x^6)^{1} &=& 90 \cdot x^{43} \\ (k=2,i=2): & \dbinom{10}{2} \dbinom{2}{2} (x^4)^{8} (x^5)^{0} (x^6)^{2} &=& 45 \cdot x^{44} \\ \hline (k=3,i=0): & \dbinom{10}{3} \dbinom{3}{0} (x^4)^{7} (x^5)^{3} (x^6)^{0} &=& 120 \cdot x^{43} \\ (k=3,i=1): & \dbinom{10}{3} \dbinom{3}{1} (x^4)^{7} (x^5)^{2} (x^6)^{1} &=& 360 \cdot x^{44} \\ (k=3,i=2): & \dbinom{10}{3} \dbinom{3}{2} (x^4)^{7} (x^5)^{1} (x^6)^{2} &=& 360 \cdot x^{45} \\ (k=3,i=3): & \dbinom{10}{3} \dbinom{3}{3} (x^4)^{7} (x^5)^{0} (x^6)^{3} &=& 120 \cdot x^{46} \\ \cdots \end{array}$$

$$\small{ \begin{array}{l||r|r|r|r|r|r|r|} &x^{40} & \\ & & 10 \cdot x^{41} & 10 \cdot x^{42} &\\ & & & 45 \cdot x^{42} & 90 \cdot x^{43} & 45 \cdot x^{44} & \\ & & & & 120 \cdot x^{43} & 360 \cdot x^{44} & 360 \cdot x^{45} & 120 \cdot x^{46}\\ & & & & & \cdots & \cdots & \cdots\\ \hline sum & x^{40} & 10\cdot x^{41} & 55\cdot x^{42} & 210 \cdot x^{43} & \cdots & \cdots & \cdots \end{array} }$$

$$\begin{array}{rcll} && [~ \frac14 \cdot x^4 + \frac14 \cdot x^5 + \frac14 \cdot x^6 ~]^{10} \\ &=& \frac{1}{1048576}\cdot x^{40} + \frac{10}{1048576}\cdot x^{41} + \frac{55}{1048576}\cdot x^{42} + \frac{210}{1048576}\cdot x^{43} + \frac{615}{1048576}\cdot x^{44}\\ &+& \frac{1452}{1048576}\cdot x^{45} + \frac{2850}{1048576}\cdot x^{46} + \frac{4740}{1048576}\cdot x^{47} + \frac{6765}{1048576}\cdot x^{48} + \frac{8350}{1048576}\cdot x^{49}\\ &+& \frac{8953}{1048576}\cdot x^{50} + \frac{8350}{1048576}\cdot x^{51} + \frac{6765}{1048576}\cdot x^{52} + \frac{4740}{1048576}\cdot x^{53} + \frac{2850}{1048576}\cdot x^{54}\\ &+& \frac{1452}{1048576}\cdot x^{55} + \frac{615}{1048576}\cdot x^{56} + \frac{210}{1048576}\cdot x^{57} + \frac{55}{1048576}\cdot x^{58} + \frac{10}{1048576}\cdot x^{59}\\ &+& \frac{1}{1048576}\cdot x^{60} \end{array}$$

heureka  17.05.2016