Answer:
a) the common difference is 20
b) [tex]x_8=115 , x_{12}=195[/tex]
c) the common difference is -13
d) [tex]a_{12}=52, a_{15}=13[/tex]
Step-by-step explanation:
a) what is the common difference of the sequence xn
Looking at the table, we get x_3=16, x_4=36 and x_5= 56
Deterring the common difference by subtracting x_4 from x_3 we get
36-16 =20
So, the common difference is 20
b) what is x_8? what is x_12
The formula used is: [tex]x_n=x_1+(n-1)d[/tex]
We know common difference d= 20, we need to find [tex]x_1[/tex]
Using [tex]x_3=16[/tex] we can find [tex]x_1[/tex]
[tex]x_n=x_1+(n-1)d\\x_3=x_1+(3-1)d\\15=x_1+2(20)\\15=x_1+40\\x_1=15-40\\x_1=-25[/tex]
So, We have [tex]x_1 = -25[/tex]
Now finding [tex]x_8[/tex]
[tex]x_n=x_1+(n-1)d\\x_8=x_1+(8-1)d\\x_8=-25+7(20)\\x_8=-25+140\\x_8=115[/tex]
So, [tex]\mathbf{x_8=115}[/tex]
Now finding [tex]x_{12}[/tex]
[tex]x_n=x_1+(n-1)d\\x_{12}=x_1+(12-1)d\\x_{12}=-25+11(20)\\x_{12}=-25+220\\x_{12}=195[/tex]
So, [tex]\mathbf{x_{12}=195}[/tex]
c) what is the common difference of the sequence [tex]a_m[/tex]
Looking at the table, we get a_7=104, a_8=91 and a_9= 78
Deterring the common difference by subtracting a_7 from a_8 we get
91-104 =-13
So, the common difference is -13
d) what is a_12? what is a_15?
The formula used is: [tex]a_n=a_1+(n-1)d[/tex]
We know common difference d= -13, we need to find [tex]a_1[/tex]
Using [tex]a_7=104[/tex] we can find [tex]x_1[/tex]
[tex]a_n=a_1+(n-1)d\\a_7=a_1+(7-1)d\\104=a_1+7(-13)\\104=a_1-91\\a_1=104+91\\a_1=195[/tex]
So, We have [tex]a_1 = 195[/tex]
Now finding [tex]a_{12}[/tex] , put n=12
[tex]a_n=a_1+(n-1)d\\a_{12}=a_1+(12-1)d\\a_{12}=195+11(-13)\\a_{12}=195-143\\a_{12}=52[/tex]
So, [tex]\mathbf{a_{12}=52}[/tex]
Now finding [tex]a_{15}[/tex] , put n=15
[tex]a_n=a_1+(n-1)d\\a_{15}=a_1+(15-1)d\\a_{15}=195+14(-13)\\a_{15}=195-182\\a_{15}=13[/tex]
So, [tex]\mathbf{a_{15}=13}[/tex]
