a-1-1-30-a-2-1-12-a-3-1-6-a-4-7-24-a-5-7-15-a-6-7-10-a-7-1-find-a-k-

Question Number 95048 by Tony Lin last updated on 22/May/20

a_{1} = - \frac{1}{30}

a_{2} = - \frac{1}{12}

a_{3} = - \frac{1}{6}

a_{4} = - \frac{7}{24}

a_{5} = - \frac{7}{15}

a_{6} = - \frac{7}{10}

a_{7} = - 1

f i n d a_{k}

Commented by Tony Lin last updated on 22/May/20

a_{1} = \frac{1}{5} C_{4}^{5} (- \frac{1}{30})

a_{2} = \frac{1}{6} C_{4}^{6} (- \frac{1}{30})

a_{3} = \frac{1}{7} C_{4}^{7} (- \frac{1}{30})

a_{k} = \frac{1}{k + 4} C_{4}^{k + 4} (- \frac{1}{30})

= \frac{(k + 4)!}{(k + 4) 4! k!} (- \frac{1}{30})

= - \frac{(k + 1) (k + 2) (k + 3)}{720}