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Popular Algebra >

expand (1+r)^{25}

Solution

expand

(1 + r)^{25}

Solution

1 + 25 r + 300 r^{2} + 2300 r^{3} + 12650 r^{4} + 53130 r^{5} + 177100 r^{6} + 480700 r^{7} + 1081575 r^{8} + 2042975 r^{9} + 3268760 r^{10} + 4457400 r^{11} + 5200300 r^{12} + 5200300 r^{13} + 4457400 r^{14} + 3268760 r^{15} + 2042975 r^{16} + 1081575 r^{17} + 480700 r^{18} + 177100 r^{19} + 53130 r^{20} + 12650 r^{21} + 2300 r^{22} + 300 r^{23} + 25 r^{24} + r^{25}

Solution steps

(1 + r)^{25}

Apply binomial theorem:

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

a = 1, b = r

= i = 0 \sum 25 (i 25) \cdot 1^{(25 - i)} r^{i}

Expand summation

= \frac{25 !}{0 ! ( 25 - 0 ) !} \cdot 1^{25} r^{0} + \frac{25 !}{1 ! ( 25 - 1 ) !} \cdot 1^{24} r^{1} + \frac{25 !}{2 ! ( 25 - 2 ) !} \cdot 1^{23} r^{2} + \frac{25 !}{3 ! ( 25 - 3 ) !} \cdot 1^{22} r^{3} + \frac{25 !}{4 ! ( 25 - 4 ) !} \cdot 1^{21} r^{4} + \frac{25 !}{5 ! ( 25 - 5 ) !} \cdot 1^{20} r^{5} + \frac{25 !}{6 ! ( 25 - 6 ) !} \cdot 1^{19} r^{6} + \frac{25 !}{7 ! ( 25 - 7 ) !} \cdot 1^{18} r^{7} + \frac{25 !}{8 ! ( 25 - 8 ) !} \cdot 1^{17} r^{8} + \frac{25 !}{9 ! ( 25 - 9 ) !} \cdot 1^{16} r^{9} + \frac{25 !}{10 ! ( 25 - 10 ) !} \cdot 1^{15} r^{10} + \frac{25 !}{11 ! ( 25 - 11 ) !} \cdot 1^{14} r^{11} + \frac{25 !}{12 ! ( 25 - 12 ) !} \cdot 1^{13} r^{12} + \frac{25 !}{13 ! ( 25 - 13 ) !} \cdot 1^{12} r^{13} + \frac{25 !}{14 ! ( 25 - 14 ) !} \cdot 1^{11} r^{14} + \frac{25 !}{15 ! ( 25 - 15 ) !} \cdot 1^{10} r^{15} + \frac{25 !}{16 ! ( 25 - 16 ) !} \cdot 1^{9} r^{16} + \frac{25 !}{17 ! ( 25 - 17 ) !} \cdot 1^{8} r^{17} + \frac{25 !}{18 ! ( 25 - 18 ) !} \cdot 1^{7} r^{18} + \frac{25 !}{19 ! ( 25 - 19 ) !} \cdot 1^{6} r^{19} + \frac{25 !}{20 ! ( 25 - 20 ) !} \cdot 1^{5} r^{20} + \frac{25 !}{21 ! ( 25 - 21 ) !} \cdot 1^{4} r^{21} + \frac{25 !}{22 ! ( 25 - 22 ) !} \cdot 1^{3} r^{22} + \frac{25 !}{23 ! ( 25 - 23 ) !} \cdot 1^{2} r^{23} + \frac{25 !}{24 ! ( 25 - 24 ) !} \cdot 1^{1} r^{24} + \frac{25 !}{25 ! ( 25 - 25 ) !} \cdot 1^{0} r^{25}

Simplify

\frac{25 !}{0 ! ( 25 - 0 ) !} \cdot 1^{25} r^{0} : 1

Simplify

\frac{25 !}{1 ! ( 25 - 1 ) !} \cdot 1^{24} r^{1} : 25 r

Simplify

\frac{25 !}{2 ! ( 25 - 2 ) !} \cdot 1^{23} r^{2} : 300 r^{2}

Simplify

\frac{25 !}{3 ! ( 25 - 3 ) !} \cdot 1^{22} r^{3} : 2300 r^{3}

Simplify

\frac{25 !}{4 ! ( 25 - 4 ) !} \cdot 1^{21} r^{4} : 12650 r^{4}

Simplify

\frac{25 !}{5 ! ( 25 - 5 ) !} \cdot 1^{20} r^{5} : 53130 r^{5}

Simplify

\frac{25 !}{6 ! ( 25 - 6 ) !} \cdot 1^{19} r^{6} : 177100 r^{6}

Simplify

\frac{25 !}{7 ! ( 25 - 7 ) !} \cdot 1^{18} r^{7} : 480700 r^{7}

Simplify

\frac{25 !}{8 ! ( 25 - 8 ) !} \cdot 1^{17} r^{8} : 1081575 r^{8}

Simplify

\frac{25 !}{9 ! ( 25 - 9 ) !} \cdot 1^{16} r^{9} : 2042975 r^{9}

Simplify

\frac{25 !}{10 ! ( 25 - 10 ) !} \cdot 1^{15} r^{10} : 3268760 r^{10}

Simplify

\frac{25 !}{11 ! ( 25 - 11 ) !} \cdot 1^{14} r^{11} : 4457400 r^{11}

Simplify

\frac{25 !}{12 ! ( 25 - 12 ) !} \cdot 1^{13} r^{12} : 5200300 r^{12}

Simplify

\frac{25 !}{13 ! ( 25 - 13 ) !} \cdot 1^{12} r^{13} : 5200300 r^{13}

Simplify

\frac{25 !}{14 ! ( 25 - 14 ) !} \cdot 1^{11} r^{14} : 4457400 r^{14}

Simplify

\frac{25 !}{15 ! ( 25 - 15 ) !} \cdot 1^{10} r^{15} : 3268760 r^{15}

Simplify

\frac{25 !}{16 ! ( 25 - 16 ) !} \cdot 1^{9} r^{16} : 2042975 r^{16}

Simplify

\frac{25 !}{17 ! ( 25 - 17 ) !} \cdot 1^{8} r^{17} : 1081575 r^{17}

Simplify

\frac{25 !}{18 ! ( 25 - 18 ) !} \cdot 1^{7} r^{18} : 480700 r^{18}

Simplify

\frac{25 !}{19 ! ( 25 - 19 ) !} \cdot 1^{6} r^{19} : 177100 r^{19}

Simplify

\frac{25 !}{20 ! ( 25 - 20 ) !} \cdot 1^{5} r^{20} : 53130 r^{20}

Simplify

\frac{25 !}{21 ! ( 25 - 21 ) !} \cdot 1^{4} r^{21} : 12650 r^{21}

Simplify

\frac{25 !}{22 ! ( 25 - 22 ) !} \cdot 1^{3} r^{22} : 2300 r^{22}

Simplify

\frac{25 !}{23 ! ( 25 - 23 ) !} \cdot 1^{2} r^{23} : 300 r^{23}

Simplify

\frac{25 !}{24 ! ( 25 - 24 ) !} \cdot 1^{1} r^{24} : 25 r^{24}

Simplify

\frac{25 !}{25 ! ( 25 - 25 ) !} \cdot 1^{0} r^{25} : r^{25}

= 1 + 25 r + 300 r^{2} + 2300 r^{3} + 12650 r^{4} + 53130 r^{5} + 177100 r^{6} + 480700 r^{7} + 1081575 r^{8} + 2042975 r^{9} + 3268760 r^{10} + 4457400 r^{11} + 5200300 r^{12} + 5200300 r^{13} + 4457400 r^{14} + 3268760 r^{15} + 2042975 r^{16} + 1081575 r^{17} + 480700 r^{18} + 177100 r^{19} + 53130 r^{20} + 12650 r^{21} + 2300 r^{22} + 300 r^{23} + 25 r^{24} + r^{25}

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