Tính nhanh:1/2+1/6+1/18+1/54+1/162+1/486

Question

minhnguyet · Answer

Đáp án+Giải thích các bước giải:

Ta có:

$2 = 2.1 = {2.3}^{0}$

$6 = 2.3 = {2.3}^{1}$

$18 = 2.9 = {2.3}^{2}$

$54 = 2.27 = {2.3}^{3}$

$162 = 2.81 = {2.3}^{4}$

$486 = 2.243 = {2.3}^{5}$

Vậy ta tính tổng

$A = \frac{1}{2} + \frac{1}{6} + \frac{1}{18} + \frac{1}{54} + \frac{1}{162} + \frac{1}{486}$

$= \frac{1}{2} (1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243})$

Ta tính

$B = 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243}$

Khi đó

$\frac{1}{3} B = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729}$

Khi đó

$B - \frac{1}{3} B = 1 - \frac{1}{729}$

$< - > \frac{2}{3} B = \frac{728}{729}$

$< - > B = \frac{364}{243}$

Suy ra

$A = \frac{1}{2} B = \frac{182}{243}$