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Popular >

expand (1+x/2)^{14}

Solution

expand

(1 + \frac{x}{2})^{14}

Solution

1 + 7 x + \frac{91 x ^{2}}{4} + \frac{91 x ^{3}}{2} + \frac{1001 x ^{4}}{16} + \frac{1001 x ^{5}}{16} + \frac{3003 x ^{6}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{8}}{256} + \frac{1001 x ^{9}}{256} + \frac{1001 x ^{10}}{1024} + \frac{91 x ^{11}}{512} + \frac{91 x ^{12}}{4096} + \frac{7 x ^{13}}{4096} + \frac{x ^{14}}{16384}

Solution steps

(1 + \frac{x}{2})^{14}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = \frac{x}{2}

= i = 0 \sum 14 (i 14) \cdot 1^{(14 - i)} (\frac{x}{2})^{i}

Expand summation

= \frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 1^{14} (\frac{x}{2})^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 1^{13} (\frac{x}{2})^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 1^{12} (\frac{x}{2})^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 1^{11} (\frac{x}{2})^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 1^{10} (\frac{x}{2})^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 1^{9} (\frac{x}{2})^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 1^{8} (\frac{x}{2})^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 1^{7} (\frac{x}{2})^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 1^{6} (\frac{x}{2})^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 1^{5} (\frac{x}{2})^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 1^{4} (\frac{x}{2})^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 1^{3} (\frac{x}{2})^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 1^{2} (\frac{x}{2})^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 1^{1} (\frac{x}{2})^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 1^{0} (\frac{x}{2})^{14}

Simplify

\frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 1^{14} (\frac{x}{2})^{0} : 1

Simplify

\frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 1^{13} (\frac{x}{2})^{1} : 7 x

Simplify

\frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 1^{12} (\frac{x}{2})^{2} : \frac{91 x ^{2}}{4}

Simplify

\frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 1^{11} (\frac{x}{2})^{3} : \frac{91 x ^{3}}{2}

Simplify

\frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 1^{10} (\frac{x}{2})^{4} : \frac{1001 x ^{4}}{16}

Simplify

\frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 1^{9} (\frac{x}{2})^{5} : \frac{1001 x ^{5}}{16}

Simplify

\frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 1^{8} (\frac{x}{2})^{6} : \frac{3003 x ^{6}}{64}

Simplify

\frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 1^{7} (\frac{x}{2})^{7} : \frac{429 x ^{7}}{16}

Simplify

\frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 1^{6} (\frac{x}{2})^{8} : \frac{3003 x ^{8}}{256}

Simplify

\frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 1^{5} (\frac{x}{2})^{9} : \frac{1001 x ^{9}}{256}

Simplify

\frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 1^{4} (\frac{x}{2})^{10} : \frac{1001 x ^{10}}{1024}

Simplify

\frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 1^{3} (\frac{x}{2})^{11} : \frac{91 x ^{11}}{512}

Simplify

\frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 1^{2} (\frac{x}{2})^{12} : \frac{91 x ^{12}}{4096}

Simplify

\frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 1^{1} (\frac{x}{2})^{13} : \frac{7 x ^{13}}{4096}

Simplify

\frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 1^{0} (\frac{x}{2})^{14} : \frac{x ^{14}}{16384}

= 1 + 7 x + \frac{91 x ^{2}}{4} + \frac{91 x ^{3}}{2} + \frac{1001 x ^{4}}{16} + \frac{1001 x ^{5}}{16} + \frac{3003 x ^{6}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{8}}{256} + \frac{1001 x ^{9}}{256} + \frac{1001 x ^{10}}{1024} + \frac{91 x ^{11}}{512} + \frac{91 x ^{12}}{4096} + \frac{7 x ^{13}}{4096} + \frac{x ^{14}}{16384}

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