|x+1|+|x+2|+...+|x+100|=101x
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1. cho f(x)=x8-101x7+101x6-101x5+...+101x2-101x+25. tính f(100)
f(x)=x8-101x7+101x6-101x5+...+101x2-101x+25
=x8-(100+1)x7+(100+1)x6-(100+1)x5+...+(100+1)x2-(100+1)x+25
f(100 ) hay x= 100
Thay 100 = x ,có :
=x8-(x+1)x7+(x+1)x6-(x+1)x5+...+(x+1)x2-(x+1)x+25
= x8 - x8 - x7+ x7 + x6 - x6 - x5 + x5 + .......................+ x3 + x2 - x2 + x + 25
= x+ 25
f(100 0 = 100 + 25 = 125
Vậy f(100 ) =125
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Ta có x = 100
=> x + 1 = 101
Khi đó A = x15 - 101x14 + 101x13 - 101x12 + ... + 101x3 - 101x2 + 101x + 2020
= x15 - (x + 1)x14 + (x + 1)x13 - (x + 1)x12 + ... + (x + 1)x3 - (x + 1)x2 + (x + 1)x + 2020
= x15 - x15 - x14 + x14 + x13 - x13 - x12 + ... + x4 + x3 - x3 - x2 + x2 + x + 2020
= x + 2020
= 101 + 2020 (Vì x = 100)
= 2121
Vậy A = 2121 khi x = 100
A = x15 - 101x14 + 101x13 - ... - 101x2 + 101x + 2020 tại x = 100
x = 100 => 101 = x + 1
Thế vào A ta được
A = x15 - ( x + 1 )x14 + ( x + 1 )x13 - ... - ( x + 1 )x2 + ( x + 1 )x + 2020
= x15 - ( x15 + x14 ) + ( x14 + x13 ) - ... - ( x3 + x2 ) + ( x2 + x ) + 2020
= x15 - x15 - x14 + x14 + x13 - ... - x3 - x2 + x2 + x + 2020
= x + 2020
= 100 + 2020 = 2120
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f(100)=x8-(100+1)x7+(100+1)x6-(100+1)x5+....+(100+1)x2-(100+1)x+25
=x8-(x+1)x7+(x+1)x6-(x+1)x5+....+(x+1)x2-(x+1)x+25
=x8-x8-x7+x7+x6-x6-x5+...+x3+x2-x2-x+25
=25
vậy f(100)=25
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Ta có: 101 = 100+1=x+1
Khi đó :
\(f\left(x\right)=x^8-101x^7+101x^6-101x^5+...+101x^2-101x+25\)
\(f\left(x\right)=x^8-\left(x+1\right)x^7+\left(x+1\right)x^6-\left(x+1\right)x^5+.....+\left(x+1\right)x^2-\left(x+1\right)x+25\)
\(f\left(x\right)=x^8-x^8-x^7+x^7+x^6-x^6+x^5+...+x^3+x^2-x^2-x+25\)
\(f\left(x\right)=-x+25\)
Vậy \(f\left(100\right)=-100+25=-75\)