CM : \(\frac{1}{\left(\sqrt{n}+\sqrt{n+1}\right)}=\sqrt{n+1}-\sqrt{n}\) (nhân chéo lên ta thấy đpcm) 

áp dụng cho S ta được:

\(\Rightarrow S=\sqrt{2}-\sqrt{2}+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}\)

\(S=\sqrt{100}-\sqrt{1}\)

S = 10 - 1 = 9 = 3^2 là số chính phương

tưởng sao,bảo nam trần cx copy bài của ng khác trên olm ==

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

cách 1:CM\(\frac{1}{\left(\sqrt{n}+\sqrt{n+1}\right)}=\sqrt{n+1}-\sqrt{n}\) (Nhân chéo lên ta thấy đpcm) 

áp dụng cho S ta được:

=>S = \(\sqrt{2}-\sqrt{2}+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}\)

S = \(\sqrt{100}-\sqrt{1}\)

S = 10 - 1 = 9 = 3² là SCP

cách 2 mình quên mất rùi sr

phần a:

nhóm S thành 50 nhóm mỗi nhóm 2 số ta có:

\(S=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{99}\left(1+2\right)=3\left(2+2^3+...+2^{99}\right)\)

Nhóm biểu thức trong ngoặc thành 25 nhóm mỗi nhóm 2 số ta có:

\(\Rightarrow S=3\left[2\left(1+2^2\right)+2^5\left(1+2^2\right)+...+2^{97}\left(1+2^2\right)\right]\)

\(\Rightarrow S=15\left(2+2^5+...+12^{97}\right)⋮15\)

phẫn c : 

ta có : S=2^1+2^2+...+2^100

2S=4+2^1+2^2+...+2^99

2S-S=(4+2^1+2^2+...+2^99)-(2^1+2^2+...+2^100)

S= 4-2^100

phẫn b : 

ta có : 2¹⁰⁰=2^3x333+1

          =(2³)³³³+2¹

         =(...8)³³³+2

         =(...8)+2=(...0)

S=4-(...0)

=>S=(...4)

dhsrtdytlitytctycvryrtvftitftftfgfjtftdtc ft ft ftf tf tftf f ft t

Vì \(\dfrac{1}{a}\left(a>1\right)< 1với\forall a\)

mà \(2^2;3^2;.....;100^2>1\)

\(=>\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{100^2}< 1\)

Đặt :

\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+......+\dfrac{1}{100^2}\)

Ta có :

\(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)

\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)

.................

\(\dfrac{1}{100^2}< \dfrac{1}{99.100}\)

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+....+\dfrac{1}{100^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+....+\dfrac{1}{99.100}\)

\(\Leftrightarrow A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+....+\dfrac{1}{99}-\dfrac{1}{100}\)

\(\Leftrightarrow A< 1-\dfrac{1}{100}< 1\left(đpcm\right)\)

⇔122+132+....+11002<11.2+12.3+....+