A = 1.2 + 2.3 + 3.4 + ... + 2017.2018

⇒ 3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 2017.218.(2019 - 2016)

= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 2017.2018.2019 - 2016.2017.2018

= 2017.2018.2019

= 2017.2018.2019

B = 2018³/3 ⇒ 3B = 2018³

Ta có:

2017.2019 = (2018 - 1).(2018 + 1)

= 2018² - 1²

= 2018.2018 - 1 < 2018.2018

⇒ 2017.2018.2019 < 2018.2018.2018

⇒ 3A < 3B

⇒ A < B

Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

Ta có

\(VT:\frac{a^{2018}+c^{2018}}{b^{2018}+d^{2018}}=\frac{b^{2018}\cdot k^{2018}+d^{2018}\cdot k^{2018}}{b^{2018}+d^{2018}}=\frac{k^{2018}\left(b^{2018}+d^{2018}\right)}{b^{2018}+d^{2018}}=k^{2018}\)

\(VP:\frac{\left(a+c\right)^{2018}}{\left(b+d\right)^{2018}}=\frac{\left(bk+dk\right)^{2018}}{\left(b+d\right)^{2018}}=\frac{k^{2018}\cdot\left(b+d\right)^{2018}}{\left(b+d\right)^{2018}}=k^{2018}\)

\(\Rightarrow VT=VP\)

Hay \(\frac{a^{2018}+c^{2018}}{b^{2018}+d^{2018}}=\frac{\left(a+c\right)^{2018}}{\left(b+d\right)^{2018}}\left(đpcm\right)\)

Lời giải:

Quy nạp. Ta chứng minh tổng quát rằng \(a^k+b^k=x^k+y^k(*)\) với \(k\in\mathbb{N}\)

Với $k=1,k=2$: hiển nhiên theo giả thiết.

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

Giả sử điều \((*)\) đúng tới $k=n$. Ta sẽ chứng minh nó cũng đúng với $k=n+1$. Tức là \(a^{n+1}+b^{n+1}=x^{n+1}+y^{n+1}\)

Thật vậy:

\(a^{n+1}+b^{n+1}=(a^n+b^n)(a+b)-a^nb-ab^n\)

\(=(x^n+y^n)(x+y)-ab(a^{n-1}+b^{n-1})\)

\(=(x^n+y^n)(x+y)-ab(x^{n-1}+y^{n-1})\)

Vì \(a^2+b^2=x^2+y^2\Rightarrow (a+b)^2-2ab=(x+y)^2-2xy\)

Mà $a+b=x+y$ nên \(2ab=2xy\Rightarrow ab=xy\)

\(\Rightarrow a^{n+1}+b^{n+1}=(x^n+y^n)(x+y)-xy(x^{n-1}+y^{n-1})=x^{n+1}+y^{n+1}\)

Quy nạp hoàn thành. Ta luôn có $(*)$. Thay $k=2018$ ta có đpcm.

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\) (1)

a) Từ (1) ta có:

\(\dfrac{a}{a+b}=\dfrac{bk}{bk+b}=\dfrac{bk}{b\left(k+1\right)}=\dfrac{k}{k+1}\) (2)

\(\dfrac{c}{c+d}=\dfrac{dk}{dk+d}=\dfrac{dk}{d\left(k+1\right)}=\dfrac{k}{k+1}\) (3)

Từ (2) và (3) suy ra \(\dfrac{a}{a+b}=\dfrac{c}{c+d}\)

b) Từ (1) ta có:

\(\dfrac{a^{2018}+c^{2018}}{b^{2018}+d^{2018}}=\dfrac{b^{2018}.k^{2018}+d^{2018}.k^{2018}}{b^{2018}+d^{2018}}=\dfrac{k^{2018}\left(b^{2018}+d^{2018}\right)}{b^{2018}+d^{2018}}=k^{2018}\) (4)

\(\dfrac{\left(a+c\right)^{2018}}{\left(b+d\right)^{2018}}=\dfrac{\left(bk+dk\right)^{2018}}{\left(b+d\right)^{2018}}=\dfrac{\left[k\left(b+d\right)\right]^{2018}}{\left(b+d\right)^{2018}}=k^{2018}\) (5)

Từ (4) và (5) suy ra \(\dfrac{a^{2018}+c^{2018}}{b^{2018}+d^{2018}}=\dfrac{\left(a+c\right)^{2018}}{\left(b+d\right)^{2018}}\)

\(a< b\Rightarrow2019a< 2019b\Rightarrow-2019a>-2019b\)

Lại có 2020 > 2018 nên \(2020-2019a>2018-2019b\).

Đáp án C