Question

a) A = 1.3 + 2.4 + 3.5 +...+ 97.99 + 98.100

b) B = 1.2.3 + 2.3.4 + 3.4.5 +...+ 48.49.50

c) C = 1/2 + 1/2^2 + 1/2^3 +...+ 1/2^10

Answer 1

a) A = 1.3 +2.4 + 3.5 +...+ 97.99 + 98.100

A = 1(2 + 1) + 2(3+1) + 3(4 + 1) +...+ 98(99+1)

= (1.2 + 2.3 + 3.4 +...+ 98.99) + (1 + 2 + 3 +...+ 98)

= [ 1.2.3 + 2.3.(4-1) +...+ 98.99.(100-97)] + [ 1.2 + 2.(3-1) + 3.(4-2) +... 98.(99-97)]

= [ 1.2.3 + 2.3.(4-1) - 1.2.3 + 3.4.(5-2) - 2.3.(4-1) +...+ 98.99.(100-97) - 97.98(99-96)] + [ 1.2 + 2.(3-1) - 1.2 + 3.(4-2) - 2.(3-1) +...+ 98.(99-97) - 97(98-96)]

= 98.99.100:3 + 98.99:2 = 323 400 + 4581 = 328251

Answer 2

b) B = 1.2.3 + 2.3.4 + 3.4.5 +...+ 48.49.50

4B = 1.2.3.4 + 2.3.4.(5-1) + 3.4.5.(6-2) +...+ 48.49.50.(51-47)

4B-B = 1.2.3.4 + 2.3.4.(5-1) - 1.2.3.4 + 3.4.5.(6-2) - 2.3.4.(5-1) +...+ 48.49.50.(51-47) - 47.48.49.(50-46)

= 48.49.50.51:4 = 1499400

Answer 3

c) C = 1/2 + 1/2^2 + 1/2^3 +...+ 1/2^10

=> 2C = 1 + 1/2 + 1/2^2 +...+ 1/2^9 (1)

= 1/2 + 1/2^2 +...+ 1/2^10

Lấy (1)-(2) ta được:

A = 1-1/2^10 = 1024/1024 - 1/1024 = 1023/1024

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