Question

যদি $a:b=b:c$ হয়, তবে প্রমাণ কর যে, $a^2b^2c^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)$$=a^3+b^3+c^3$

Answer 1

দেওয়া আছে,
$a:b=b:c$ বা, $\frac{a}{b}=\frac{b}{c}$ বা, $b^2=ac$

Left Hand Side,
$a^2b^2c^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)$

$=\frac{a^2b^2c^2}{a^3}+\frac{a^2b^2c^2}{b^3}+\frac{a^2b^2c^2}{c^3}$

$=\frac{b^2c^2}{a}+\frac{a^2c^2}{b}+\frac{a^2b^2}{c}$

$=\frac{b^2c^2}{a}+\frac{\left(ac\right)^2}{b}+\frac{a^2b^2}{c}$

$=\frac{ac\cdot c^2}{a}+\frac{(b^2)^2}{b}+\frac{a^2\cdot ac}{c}$

[$\because b^2=ac$]

$=\frac{ac^3}{a}+\frac{b^4}{b}+\frac{a^3c}{c}$

$=c^3+b^3+a^3$

$=a^3+b^3+c^3$

$=$ Right Hand Side

$\therefore$ Left Hand Side $=$ Right Hand Side

অর্থাৎ, $a^2b^2c^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)$$=a^3+b^3+c^3$