If `_r=|2^r 2.3^r-1 4.5^r-1alphabetagamma2^n-1 3^n-1 5^n-1|` , then find the value of `dot`

Answer

0

Explanation

`overset(n)underset(r=1)(Sigma)2^(r-1)=1+2+2^(2)+.......+2^(n+1)=1xx((2^(n)-1))/(2-1)=2^(n)-1`
`overset(n)underset(r=1)(Sigma)2xx3^(r-1)=2(1+3+3^(2)+...+3^(n-1))=(2(3^(n)-1))/(3-1)=3^(n)-1`
`overset(n)underset(r=1)(Sigma)4xx5^(r-1)=4(1+5+5^(2)+....+5^(n)-1)=(4(5^(n)-1))/(5-1)=5^(n)-1`
`:. overset(n)underset(r=1)(Sigma)Delta_(r ) =|{:(2^(n-1),,3^(n-1),,5^(n-1)),(alpha,,beta,,gamma),(2^(n-1),,3^(n-1),,5^(n-1)):}|=0`