The value of the expression ((1 ⋅ 2 ⋅ 4+2 ⋅ 4 ⋅ 8+ ldots ldots ldots+n ⋅ 2 n ⋅ 4 n/1 ⋅ 3 ⋅ 9+2 ⋅ 6 ⋅ 18+ ldots ldots ldots+n ⋅ 3 n ⋅ 9 n))(1/3), is

Question

The value of the expression ((1 ⋅ 2 ⋅ 4+2 ⋅ 4 ⋅ 8+ ldots ldots ldots+n ⋅ 2 n ⋅ 4 n/1 ⋅ 3 ⋅ 9+2 ⋅ 6 ⋅ 18+ ldots ldots ldots+n ⋅ 3 n ⋅ 9 n))(1/3), is (A) (1/3) (B) (2/3) (C) (4/9) (D) (8/27)

Tardigrade Admin · Accepted Answer

(( displaystyle∑r=1n 8 r3/ displaystyle∑r=1n 27 r3))(1/3)=((8/27))1 / 3=(2/3) .

Q. The value of the expression $(\frac{1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + \dots\dots\dots + n \cdot 2 n \cdot 4 n}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + \dots\dots\dots + n \cdot 3 n \cdot 9 n})^{\frac{1}{3}}$ , is

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{9}$

$\frac{8}{27}$

$⎝ ⎛ \frac{r = 1 \sum n 8 r ^{3}}{r = 1 \sum n 27 r ^{3}} ⎠ ⎞^{\frac{1}{3}} = (\frac{8}{27})^{1/3} = \frac{2}{3} .$

Q. The value of the expression (1⋅3⋅9+2⋅6⋅18+………+n⋅3n⋅9n1⋅2⋅4+2⋅4⋅8+………+n⋅2n⋅4n​)31​, is

31​

32​

94​

278​