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Popular Álgebra >

expandir (1+r)^{10}

Solução

expandir

(1 + r)^{10}

Solução

1 + 10 r + 45 r^{2} + 120 r^{3} + 210 r^{4} + 252 r^{5} + 210 r^{6} + 120 r^{7} + 45 r^{8} + 10 r^{9} + r^{10}

Passos da solução

(1 + r)^{10}

Aplicar o teorema do binômio:

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

a = 1, b = r

= i = 0 \sum 10 (i 10) \cdot 1^{(10 - i)} r^{i}

Expandir a soma

= \frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} r^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} r^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} r^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} r^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} r^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} r^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} r^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} r^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} r^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} r^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} r^{10}

Simplificar

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} r^{0} : 1

Simplificar

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} r^{1} : 10 r

Simplificar

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} r^{2} : 45 r^{2}

Simplificar

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} r^{3} : 120 r^{3}

Simplificar

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} r^{4} : 210 r^{4}

Simplificar

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} r^{5} : 252 r^{5}

Simplificar

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} r^{6} : 210 r^{6}

Simplificar

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} r^{7} : 120 r^{7}

Simplificar

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} r^{8} : 45 r^{8}

Simplificar

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} r^{9} : 10 r^{9}

Simplificar

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} r^{10} : r^{10}

= 1 + 10 r + 45 r^{2} + 120 r^{3} + 210 r^{4} + 252 r^{5} + 210 r^{6} + 120 r^{7} + 45 r^{8} + 10 r^{9} + r^{10}

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