Question

Tính tổng sau :

a ) A = 2 + 2^2 + 2^3 + ......... + 2^100

b ) B = 1 + 3 + 3^2  + ......... + 3^255

c ) C = 1 + 4 + 4^2 +........+ 4^100

d ) D = 1 + 5 + 5^2 + ........... 5^1000

( giải chi tiết cho mk nhé ! )

Answer 1

a) A = 2 + 2² + 2³ + ... + 2¹⁰⁰

2A = 2² + 2³ + 2⁴ + ... + 2¹⁰¹

2A - A = (2² + 2³ + 2⁴ + ... + 2¹⁰¹) - (2 + 2² + 2³ + ... + 2¹⁰⁰)

A = 2¹⁰¹ - 2

b) B = 1 + 3 + 3² + ... + 3²⁵⁵

3B = 3 + 3² + 3³ + ... + 3²⁵⁶

3B - B = (3 + 3² + 3³ + ... + 3²⁵⁶) - (1 + 3 + 3² + ... + 3²⁵⁵)

2B = 3²⁵⁶ - 1

B = \(\frac{3^{256}-1}{2}\)

c) C = 1 + 4 + 4² + ... + 4¹⁰⁰

4C = 4 + 4² + 4³ + ... + 4¹⁰¹

4C - C = (4 + 4² + 4³ + ... + 4¹⁰¹) - (1 + 4 + 4² + ... + 4¹⁰⁰)

3C = 4¹⁰¹ - 1

C = \(\frac{4^{101}-1}{3}\)

d) D = 1 + 5 + 5² + ... + 5¹⁰⁰⁰

5D = 5 + 5² + 5³ + ... + 5¹⁰⁰¹

5D - D = (5 + 5² + 5³ + ... + 5¹⁰⁰¹) - (1 + 5 + 5² + ... + 5¹⁰⁰⁰)

4D = 5¹⁰⁰¹ - 1

D = \(\frac{5^{1001}-1}{4}\)