Find-the-sum-30-C-0-5-100-30-C-1-5-98-3-3-30-C-2-5-96-3-6-

Question Number 48795 by rahul 19 last updated on 28/Nov/18

F i n d t h e s u m :

^{30} C_{0} . 5^{100} -^{30} C_{1} . 5^{98} . 3^{3} +^{30} C_{2} . 5^{96} . 3^{6} - \dots = ?

Commented by rahul 19 last updated on 29/Nov/18

T_{r} = \underset{r = 0}{\sum^{30}} {(- 1)}^{r 30} C_{r} . {(5)}^{- 2 r + 100} . {(3)}^{3 r}

T_{r} = \underset{r = 0}{\sum^{30}}^{30} C_{r} {(25)}^{- r + 50} {(- 27)}^{r}

T_{r} = {(25)}^{20} \underset{r = 0}{\sum^{30}}^{30} C_{r} {(25)}^{30 - r} {(- 27)}^{r}

T_{r} = {(25)}^{20} {(25 - 27)}^{30}

T_{r} = 2^{30} 5^{40} .

Answered by ajfour last updated on 28/Nov/18

= {(25)}^{20} [^{30} C_{0} {(25)}^{30} -^{30} C_{1} {(25)}^{29} (27) + . .]

= {(25)}^{20} {(25 - 27)}^{30}

= 2^{30} 5^{40} .

Commented by rahul 19 last updated on 29/Nov/18

thank you sir!��