Question

bài 1 :1.2 + 2.3 + 3.4 + ... + 201. 202

bài 2 : s = 4 . 5 + 5.6 + 6.7+...+ 100.101

bài 3 : s= 1.2.3 + 2.3.4+ 3.4.5+...+ 98 . 99 .100

bài 4 : tính tổng sau: 1/5 + 1/25 + ... + 1/5 mu 100

Answer 1

Bài 1:

$A=1.2+2.3+3.4+...+201.202$

$3A=1.2.3+2.3(4-1)+3.4(5-2)+....+201.202(203-200)$

$=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+....+201.202.203-200.201.202$

$=(1.2.3+2.3.4+3.4.5+...+201.202.203)-(1.2.3+2.3.4+....+200.201.202)$

$=201.202.203$

$\Rightarrow A=\frac{201.202.203}{3}=2747402$

Answer 2

Bài 2:

$S=4.5+5.6+6.7+....+100.101$

$3S=4.5(6-3)+5.6.(7-4)+6.7.(8-5)+....+100.101(102-99)$

$=4.5.6-3.4.5+5.6.7-4.5.6+6.7.8-5.6.7+....+100.101.102-99.100.101$

$=(4.5.6+5.6.7+6.7.8+...+100.101.102)-(3.4.5+4.5.6+5.6.7+...+99.100.101)$

$=100.101.102-3.4.5$

$\Rightarrow S=\frac{100.101.102-3.4.5}{3}=343380$

Answer 3

Bài 3:

$S=1.2.3+2.3.4+3.4.5+...+98.99.100$

$4S=1.2.3(4-0)+2.3.4(5-1)+3.4.5(6-2)+...+98.99.100(101-97)$

$=(1.2.3.4+2.3.4.5+3.4.5.6+...+98.99.100.101)-(0.1.2.3+1.2.3.4+2.3.4.5+...+97.98.99.100)$

$=98.99.100.101$

$\Rightarrow S=\frac{98.99.100.101}{4}$

Answer 4

Bài 4:

$S=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^{100}}$

$5S=1+\frac{1}{5}+\frac{1}{5^2}+....+\frac{1}{5^{99}}$

$\Rightarrow 5S-S=1-\frac{1}{5^{100}}$

$\Rightarrow S=\frac{1}{4}(1-\frac{1}{5^{100}})$