Question

Find an example of an arithmetic sequence that has an explicit pattern. Then determine the formula for that sequence and use it to define the given term.

Answer 1

11)

5, -5, -15, -25, ...

-5-5 = -10; -15-(-5) = -10; -25-(-15) = -10 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 10 less than previous one)

difference: d = -10

first term: a = 5

so the formula:

`a_n=a+d(n-1)\\\\a_n=5+(-10)(n-1)\\\\a_n =5-10n+10\\\\\underline{a_n=-10n+15}`

and:

`a_(20)=-10\cdot20+15=-200+15=-185`

12)

19, 26, 33, 40, ...

26-19 = 7; 33-26 = 7; 40-33 = 7 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 7 more than previous one)

difference: d = 7

first term: a = 19

so the formula:

`a_n=a+d(n-1)\\\\a_n=19+7(n-1)\\\\a_n =19+7n-7\\\\\underline{a_n=7n+12}`

and:

`a_(39)=7\cdot39+12=273+12=285`

13)

-20, -29, -38, -47, ...

-29-(-20) = -9; -38-(-28) = -9; -47-(-38) = -9 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 9 less than previous one)

difference: d = -9

first term: a = -20

so the formula:

`a_n=a+d(n-1)\\\\a_n=-20+(-9)(n-1)\\\\a_n =-20-9n+9\\\\\underline{a_n=-9n-11}`

and:

`a_(12)=-9\cdot12-11=-108-11=-119`

11)

1, -2, 3, -4, ...

-5-1 = -3; 3-(-2) = 5 ≠ -3 ⇒ this is NOT an example of arithmetic sequence

however it has explicit patern:

`a_1=1 =1\cdot\left(-1\right)^0= 1\cdot\left(-1\right)^(1-1)\\\\a_2=-2=2\cdot\left(-1\right)^1=2\cdot\left(-1\right)^(2-1)\\\\a_3=3=3\cdot\left(-1\right)^2=3\cdot\left(-1\right)^(3-1)\\\\a_4=-4=4\cdot\left(-1\right)^3=4\cdot\left(-1\right)^(4-1)\\\\\underline{a_n=n\cdot\left(-1\right)^(n-1)}`

so:

`a_(33)=33\cdot\left(-1\right)^(33-1)=33\cdot\left(-1\right)^(32)=33\cdot1=33`