1,

a,

`a_n` = `3n^2`+n-4

`a_6` = `3*6^2`+6-4 = 110

`a_(10)` = `3*10^2`+10-4 = 306

b,

`b_n` = `2^n`-3

`b_6` = `2^6`-3 = 64-3 = 61

`b_(10)` = `2^(10)`-3 = 1024-3 = 1021


2,

a,

5n-43=17

5n = 17+43=60

n = `60/5` = 12


b,

`3^n`-64 = 17

`3^n` = 17+64 = 81

n = `(log(81))/(log(3))` = 4


3,

a,

`0.5*n^3`-9 = -9,5

`0.5*n^3` = -9,5+9 = -0,5

`n^3` = -0.5/0.5 = -1

n = `root(3)(-1)` = -1

Nem tagja, n értékének pozitív egész számoknak kell lenniük.


b,

`(3n+2)/(2n)` = -9,5

3n+2 = `-9.5*2n` = -19n

22n = -2

n = `2/22` = `1/11`

Nem tagja, pozitív egésznek kell lennie.


4,

`a_1` = 3

`a_(n+1)` = 3`a_n`-2

`a_2` = `3*a_1`-2 = `3*3`-2 = 9-2 = 7

`a_3` = `3*a_2`-2 = `3*7`-2 = 19

`a_4` = `3*a_3`-2 = `3*19`-2 = 55

`a_5` = `3*a_4`-2 = `3*55`-2 = 163

`a_6` = `3*a_5`-2 = `3*163`-2 = 487


5,

`a_7` = 503 = `a_1` + 6d

`a_(33)` = 399 = `a_1` +32d

Kivonjuk egymásból a kettőt:

26d = 399-503 = -104

d = -4

Visszahelyettesítünk az egyikbe:

`a_1` + `6*(-4)` = 503

`a_1` = 503-24 = 527

A sorozat első tagja `a_1` = 527, differenciája: d= -4


6,

`a_1` + `a_2` + `a_3` = `a_1` + `a_1` + d + `a_1` + 2d = `3*a_1` + 6d = 12

osztjuk hárommal:

`a_1` + d = 4

`a_8` - `a_5` = `a_1` + 7d - (`a_1` + 4d) = 3d = 12

d = 4


`a_1` + 4 = 4

`a_1` = 0

A sorozat első tagja 0, differenciája 4.


7,

`a_1` = 7

`a_8` = 35 = `a_1` + 7d

7 + 7d = 35

7d = 28

d = 4

`a_2` = `a_1` + d = 7+4 = 11

`a_3` = 11+4 = 15

`a_4` = 15+4 = 19

`a_5` = 19+4 = 23

`a_6` = 23+4 = 27

`a_7` = 27+4 = 31



8,

`a_3` = -17,5 = `a_1` + 2d

`S_8` = `(a_1+a_8)*8/2` = `(a_1+a_1+7d)*4` = `8*a_1` +28d

`S_(9-16)` = `(a_9+a_(16))*8/2` = `(a_1+8d+a_1+15d)*4` = `8a_1` + 92d

`8*a_1` +28d = `2*(8a_1 + 92d)`

`8a_1` = 156d

`a_1` = -19,5d

`a_3` = `-19.5d`+2d = 17.5d = -17.5

d = -1

`a_1` = `a_3`-2d = -17.5+2 = -15.5

`S_(10)` = `(a_1+a_1+9d)*10/2` = `10a_1`+45d = `10*-(15.5)-45` = -200


9,

Az első háromjegyű páratlan szám a 101.

Az utolsó háromjegyű páratlan szám a 999.

Hány darab van? `(999-101)/2`+1 = 450 darab van.

`S_("páratlan 3 jegyű")` = `((101+999)*450)/2` = 247500