Question

Thu gọn :

a) A = 2¹ + 2² + 2³ + ............ + 2⁹⁹ + 2¹⁰⁰ .

b) B = 2¹ + 2³ + 2⁵ + ............. + 2²⁰¹³ .

c) C = 2¹ + 2⁴ + 2⁷ + .............. + 2¹⁰⁰³ .

d) D = 3¹ + 3² + 3³ + .............. + 3^{201 .}

Answer 1

a) Đặt A = 2¹ + 2² + 2³ +....+2⁹⁹ + 2¹⁰⁰

=> 2A = 2+ 2² + 2³ +...+2¹⁰⁰ + 2¹⁰¹

=> 2A - A = 2 + 2² + 2³ +...+2¹⁰⁰ + 2¹⁰¹ - (1+2+2²+2³+...+2⁹⁹+2¹⁰⁰)

=> A = 2 + 2² + 2³ +...+ 2¹⁰⁰ +2¹⁰¹ -1 - 2 - 2² - 2³ -...- 2⁹⁹ - 2¹⁰⁰

= 2¹⁰¹ -1

Vậy....

Answer 2

b) B = 2 + 2³ + 2⁵ + ... + 2²⁰¹³

4B = 2³ + 2⁵ + 2⁷ + ... + 2²⁰¹⁵

4B - B = (2³ + 2⁵ + 2⁷ + ... + 2²⁰¹⁵) - (2 + 2³ + 2⁵ + ... + 2²⁰¹³)

3B = 2²⁰¹⁵ - 2

B = \(\dfrac{2^{2015}-2}{3}\)

Answer 3

c) C = 2 + 2⁴ + 2⁷ + ... + 2¹⁰⁰³

8C = 2⁴ + 2⁷ + 2¹⁰ + ... + 2¹⁰⁰⁶

8C - C = (2⁴ + 2⁷ + 2¹⁰ + ... + 2¹⁰⁰⁶) - (2 + 2⁴ + 2⁷ + ... + 2¹⁰⁰³)

7C = 2¹⁰⁰⁶ - 2

C = \(\dfrac{2^{1006}-2}{7}\)