Tính giá trị các biểu thức sau:
1/ A = \(6+5^2+5^3+5^4+...+5^{1996}+5^{1997}\)
2/ B = \(10+9^2+9^3+9^4+...+9^{2004}+9^{2005}\)
3/ C = \(\dfrac{5^{997}\left(5^{100}+2\right)-10.5^{996}-1}{4}\)
4/ D = \(x^{20}-2006x^{19}+2006x^{18}-2006x^{17}+...+2006x^2-2006x+2006\) với x = 2005
5/ E =\(x^8-2007x^7+2006x^6-2006x^5+...+2000x-2000x+2000\) với x = 1999
1/ \(A=6+5^2+5^3+5^4+...+5^{1996}+5^{1997}\)
\(=1+5+5^2+5^3+5^4+...+5^{1996}+5^{1997}\)
\(\Rightarrow5A=5+5^2+5^3+5^4+5^5+...+5^{1997}+5^{1998}\)
Khi đó:
\(5A-A=\left(5+5^2+5^3+5^4+5^5+...+5^{1997}+5^{1998}\right)-\left(1+5+5^2+5^3+5^4+...+5^{1996}+5^{1997}\right)\)
\(4A=5+5^2+5^3+5^4+5^5+...+5^{1997}+5^{1998}-1-5-5^2-5^3-5^4-...-5^{1996}-5^{1997}\)
\(4A=5^{1998}-1\)
\(\Rightarrow A=\dfrac{5^{1998}-1}{4}\)
2/ \(B=10+9^2+9^3+9^4+...+9^{2004}+9^{2005}\)
\(=1+9+9^2+9^3+9^4+...+9^{2004}+9^{2005}\)
\(\Rightarrow9B=9+9^2+9^3+9^4+9^5+...+9^{2005}+9^{2006}\)
Khi đó:
\(9B-B=\left(9+9^2+9^3+9^4+9^5+...+9^{2005}+9^{2006}\right)-\left(1+9+9^2+9^3+9^4+...+9^{2004}+9^{2005}\right)\)
\(8B=9+9^2+9^3+9^4+9^5+...+9^{2005}+9^{2006}-1-9-9^2-9^3-9^4-...-9^{2004}-9^{2005}\)
\(8B=9^{2006}-1\)
\(\Rightarrow B=\dfrac{9^{2006}-1}{8}\)
3/ \(C=\dfrac{5^{997}\left(5^{100}+2\right)-10.5^{996}-1}{4}\)
\(=\dfrac{5^{997}.5^{100}+5^{997}.2-10.5^{996}-1}{4}\)
\(=\dfrac{5^{1097}+5^{996}.5.2-10.5^{996}-1}{4}\)
\(=\dfrac{5^{1097}+10.5^{996}-10.5^{996}-1}{4}\)
\(=\dfrac{5^{1097}-1}{4}\)
4/ Theo giả thiết, ta có: x = 2005 => x - 2005 = 0
Do đó:
\(D=x^{20}-2006x^{19}+2006x^{18}-2006x^{17}+...+2006x^2-2006x+2006\)
\(=\left(x^{20}-2005x^{19}\right)-\left(x^{19}-2005x^{18}\right)+\left(x^{18}-2005x^{17}\right)+...+\left(x^2-2005x\right)-\left(x-2005\right)+1\)
\(=x^{19}\left(x-2005\right)-x^{18}\left(x-2005\right)+x^{17}\left(x-2005\right)+...+x\left(x-2005\right)-\left(x-2005\right)+1\)
\(=1\)
