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`

If Delta_r =|[2^(r-1) ,2.3^(r-1), 4.5^(r-1)],[alpha,beta,gamma],[2^n-1, 3^n-1, 5^n-1]| , then find the value of Delta

if Delta_r=|[x,y,z],[2^r,2.3^r,3.4^r],[2(2^n-1),3.(3^n-1),4.(4^n-1)]| find the value of sumDelta_r

If "Delta"_r=|[2^(r-1),2. 3^(r-1),4. 5^(r-1)],[x, y ,z],[2^n-1, 3^n-1, 5^n-1]| Show that sum_(r=1)^n"Delta"_r = Constant

If Delta_r=|[2^(r-1) , 2.3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 , 3^n-1 , 5^n-1]|dot Show that sum_(r=1)^n Delta_r=Con s t a n t

If D_(r) = |(r,1,(n(n +1))/(2)),(2r -1,4,n^(2)),(2^(r -1),5,2^(n) -1)| , then the value of sum_(r=1)^(n) D_(r) , is

If sum_(r=1)^n T_r=(3^n-1), then find the sum of sum_(r=1)^n1/(T_r) .

If .^(n)C_(r-1): .^(n)C_(r): .^(n)C_(r+1)=3:4:5 , find the values of n and r.

If .^(n+1)C_(r+1.): ^nC_r: ^(n-1)C_(r-1)=11:6:3 find the values of n and r.

If ""^(n)C_(r -1) = 36, ""^(n)C_(r) = 84 " and " ""^(n)C_(r +1) = 126 , then find the value of ""^(r)C_(2) .

If .^(n+1)C_(r+1):^(n)C_(r):^(n-1)C_(r-1)=11:6:3 , find the values of n and r.