Respuesta :
hello :
the arithmetic sequence 9, –1, –11, –21 has common difference :
d = (-21)-(-11) =(-11)-(-1) = (-1)-(9) = -10
sequence is defined recursively using the formula f(n +1) = f(n) – 5 (arith -seq)
is : C) 3, –2, –7, –12, ...
because : (-12)-(-7)=(-7)-(-2)=(-2)-(3) = -5 = d (common difference)
the arithmetic sequence 9, –1, –11, –21 has common difference :
d = (-21)-(-11) =(-11)-(-1) = (-1)-(9) = -10
sequence is defined recursively using the formula f(n +1) = f(n) – 5 (arith -seq)
is : C) 3, –2, –7, –12, ...
because : (-12)-(-7)=(-7)-(-2)=(-2)-(3) = -5 = d (common difference)
Answer: The correct options are
Part A : option (a) the common difference is -10.
Part B : option (c) 3, -2, -7, -12, . . .
Step-by-step explanation: We are given to solve the following two problems
PART A :
We are given to select the statement that best describes the relationship between the successive terms in the following sequence :
9, -1, -11, -21, . . .
If a(n) denotes the n-th term of the given sequence, then we see that
[tex]a(2)-a(1)=-1-9=-10,\\\\a(3)-a(2)=-11-(-1)=-11+1=-10,\\\\a(4)-a(3)=-21-(-11)=-21+11=-10,\\\\\vdots~~~\vdots~~~\vdots[/tex]
Since the difference between any two consecutive term of the given sequence is -10, so
there is a common difference of -10.
Option (a) is CORRECT.
PART B :
Given that a sequence is defined recursively using th efollowing formula :
[tex]f(n+1)=f(n)-5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to select the sequence that could be generated using formula (i).
Let f(n) = 1. Substituting n = 1, 2, 3, 4, . . . in equation (i), we get
[tex]f(1+1)=f(1)-5\\\\\Rightarrow f(2)=1-5=-4,\\\\f(3)=f(2)-5=-4-5=-9,\\\\f(4)=f(3)-5=-9-5=-14.~~.~~.~~.[/tex]
In this case, the sequence is 1, -4, -9, -14, . . .
Let f(n) = 2. Substituting n = 1, 2, 3, 4, . . . in equation (i), we get
[tex]f(1+1)=f(1)-5\\\\\Rightarrow f(2)=2-5=-3,\\\\f(3)=f(2)-5=-3-5=-8,\\\\f(4)=f(3)-5=-8-5=-13.~~.~~.~~.[/tex]
In this case, the sequence is 2, -3, -8, -13, . . .
Let f(n) = 3. Substituting n = 1, 2, 3, 4, . . . in equation (i), we get
[tex]f(1+1)=f(1)-5\\\\\Rightarrow f(2)=3-5=-2,\\\\f(3)=f(2)-5=-2-5=-7,\\\\f(4)=f(3)-5=-7-5=-13.~~.~~.~~.[/tex]
In this case, the sequence is 3, -2, -7, -12, . . .
Let f(n) = 4. Substituting n = 1, 2, 3, 4, . . . in equation (i), we get
[tex]f(1+1)=f(1)-5\\\\\Rightarrow f(2)=4-5=-1,\\\\f(3)=f(2)-5=-1-5=-6,\\\\f(4)=f(3)-5=-6-5=-11.~~.~~.~~.[/tex]
In this case, the sequence is 4,-1, -6, -11, . . .
Thus, the correct sequence that could be generated is 3, -2, -7, -12, . . .
Option (c) is CORRECT.
