Algebra 1 benchmark

Question

Consider the following arithmetic sequence, defined recursively:

a1 = 19

An =An-1-4

Select the explicit formula for this sequence:

A: An = 19 + 4(n -1)

B: An = 19 • 4(n - 1)

C: An = 19 - 4(n - 1)

D: An = 19 • -4(n - 1)

Answer 1

This is a formula. Look up explicit formula for arithmetic sequence.

Answer 2

Terms in a sequence are identified by their number. For instance, the first term is a₁ (n = 1 in a_n) and the second term is a₂ (n = 2 in a_n). In this case the first term, also known as a₁, is 19, The equation tells us how to find the second term (a₂):

a_n = a_n-1 - 4

For n = 2:

a₂ = a_2-1 - 4

a₂ = a₁ - 4 but a₁ = 19 so:

a₂ = 19 - 4

a₂ = 15

And the equation then tells us how to find a₃:

a_n = a_n-1 - 4

For n = 3:

a₃ = a_3-1 - 4

a₃ = a₂ - 4 but a₂ = 15 so:

a₃ = 15 - 4

a₃ = 11

Try each of the equations A - D to see which one gives the same sequence 19, 15, 11 . . .

A) a_n = 19 + 4(n -1) let n = 2 to see if we get 15:

a₂ = 19 + 4(2 - 1)

a₂ = 19 + 4 = 23 No, this doesn't give us 15.

B) a_n = 19 • 4(n -1) let n = 2 to see if we get 15:

a₂ = 19 • 4(2 - 1)

a₂ = 19 • 4 = 76 No, this doesn't give us 15.

C) a_n = 19 - 4(n -1) let n = 2 to see if we get 15:

a₂ = 19 - 4(2 - 1)

a₂ = 19 - 4 = 15 GREAT! This appears to work. Lets try n = 3 just to make sure (a₃ = 11):

a₃ = 19 - 4(3 - 1)

a₃ = 19 - 4(2) = 19 - 8 = 11 GREAT! This is our answer. Try answer D just to verify it doesn't work.