a,\(D=10+100+......+1000...000-1-1-.....-1\) có 50 chữ số 0 và 50 số 1

\(=111.....111-50\) có 51 chữ số 1          \(=111.....1061\) có 48 chữ số 1

b,tương tự a

c,\(1-2^2+3^2-4^2+.......+99^2-100^2\)

\(=\left(1-2\right).\left(1+2\right)+\left(3-4\right)\left(3+4\right)+......+\left(99-100\right)\left(99+100\right)\)

\(=-\left(3+7+.....+199\right)\)\(=-\frac{\left(199+3\right).50}{2}=-5050\)

d,\(G=1.1!+2.2!+.......+100.100!\)

\(=\left(2-1\right).1!+\left(3-1\right).2!+.....+\left(101-1\right).100!\)

\(=2!-1!+3!-2!+.......+101!-100!\)

\(=101!-1!\)

Tinh tonga) D= 9+99+999+9999+...+999....9 (50 chu so 9)b) E= 9+99+999+...+999...9 (200 chu so 9)c)C=1−22+32−42+...+992−1002d) G= 1.1!+ 2. 2!+3.3!+ ... +100.100!

a) 9 + 99 + 999 + ... + 999999

= (10 - 1) + (100 - 1) + (1000 - 1) + ... + (1000000 - 1)

= (10¹ + 10² + 10³ + ... + 10⁶) - (1.6)

= 1111110 - 6 = 1111104

b) 1 + 11 + 111 + ... + 1111111

= 1 + (10¹ + 1) + (10² + 10¹ + 1) + ... + (10⁶ + 10⁵ + 10⁴ + 10³ + 10² + 10¹ + 1)

= 10¹ . 6 + 10² . 5 + 10³ . 4 + ... + 10⁶. 1) + (1 + 1.6)

= 60 + 500 + 4000 + ... + 1000000 + 7

= 1234560 + 7 = 1234567

c) C = 1.2 + 2.3 + 3.4 + 4.5 + ... + 98.99

3C = 1.2.3 + 2.3.3 + 3.4.3 + 4.5.3 + ... + 98.99.3

3C = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 98.99.(100 - 97)

3C = 1.2.3 + 2.3.4 - 2.3.1 + 3.4.5 - 3.4.2 +...+ 98.99.100 - 98.99.97

3C = 98.99.100

C = \(\dfrac{98.99.100}{3}\) = 323400

d) D = 1.3.5 + 3.5.7 + 5.7.9 + ... + 95.97.99

8D = 1.3.5.8 + 3.5.7.8 + 5.7.9.8 + ... + 95.97.99.8

8D = 1.3.5.(7 + 1) + 3.5.7.(9 - 1) + 5.7.9.(11 - 3) + ... + 95.97.99.(101 - 93)

8D = 1.3.5.7 + 1.3.5.1 + 3.5.7.9 - 3.5.7.1 + 5.7.9.11 - 5.7.9.3 + ... + 95.97.99.101 - 95.97.99.93

8D = 1.3.5.1 + 95.97.99.101

D = \(\dfrac{1.3.5.1+95.97.99.101}{8}=15517600\)

B = 9 + 99 + 999 + ... + 999...99 ( 100 chữ số 9 )

   = (10-1) + (10² -1 ) + (10³ -1) + .... + (10¹⁰⁰ -1)

   =  10 + 10² +10³ + ......+ 10¹⁰⁰ - 100 

   = 111...11 0 ( 100 chữ số 1 ; 1 chữ số 0 )               - 100

   = 1111...1 010 ( 98 chữ số 1 )