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

Options

A. xyz
B. 1
C. `-1`
D. 0

Answer

D

Explanation

Using the properties of determinants, we have
`underset(r=1)overset(n)sum Delta_(r) = |(underset(r=1)overset(n)Sigma2^(r -1),underset(r=1)overset(n)Sigma 2.3^(r-1),underset(r=1)overset(n)Sigma 4.5^(r -1)),(x,y,z),(2^(n) -1,3^(n) -1,5^(n) -1)|`
Now, `underset(r =1)overset(n)sum 2^(r-1) = 1 + 2 + 2^(2) + ..._ 2^(n-1) = 1.((2^(n) -1))/((2 -1)) = 2^(n) -1`,
`underset(r =1)overset(n)sum 2.3^(r -1)= 2[1 + 3 + 3^(2) + ...+ 3^(n-1)] = 3^(n) -1`
and, `underset(r =1)overset(n)sum 4.5^(r-1) = 4[1 + 5 + 5^(2) + ...+ 5^(n-1)] = 5^(n) -1`
`:. underset(r=1)overset(n)sum Delta_(r) = |(2^(n) -1,3^(n) -1,5^(n) -1),(x,y,z),(2^(n) -1,3^(n) -1,5^(n) -1)| =0 [ ( :' R_(1) and R_(2)),("are identical")]`