Question

look at this sequence
95, 80, 65, 50, ...
which function represents this sequence?
a. \\( a(n) = -15n + 95 \\)
b. \\( a(n) = 95n - 15 \\)
c. \\( a(n) = -15n + 110 \\)
d. \\( a(n) = 110n - 15 \\)

Explanation:

Step1: Identify the type of sequence

This is an arithmetic sequence. The general formula for an arithmetic sequence is \( a(n)=a_1+(n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

First, find the common difference \( d \). For the sequence \( 95, 80, 65, 50,\dots \), \( d=80 - 95=- 15 \), \( a_1 = 95 \).

We can also express the arithmetic sequence formula in the slope - intercept form \( y=mx + b \) (here \( a(n)=mn + b \)). The slope \( m \) is the common difference \( d=-15 \), so the function should be in the form \( a(n)=-15n + b \).

Step2: Find the value of \( b \)

We know that when \( n = 1 \), \( a(1)=95 \). Substitute \( n = 1 \) and \( m=-15 \) into \( a(n)=-15n + b \):

\( 95=-15\times1 + b \)

Solve for \( b \):

\( b=95 + 15=110 \)

So the function is \( a(n)=-15n + 110 \)

We can also check by plugging in values of \( n \):

• For option A: When \( n = 2 \), \( a(2)=-15\times2+95=-30 + 95 = 65

eq80 \), so A is wrong.

• For option B: When \( n = 2 \), \( a(2)=95\times2-15 = 190 - 15=175

eq80 \), so B is wrong.

• For option C: When \( n = 1 \), \( a(1)=-15\times1 + 110=95 \); when \( n = 2 \), \( a(2)=-15\times2+110=-30 + 110 = 80 \); when \( n = 3 \), \( a(3)=-15\times3+110=-45 + 110 = 65 \); when \( n = 4 \), \( a(4)=-15\times4+110=-60 + 110 = 50 \), which matches the sequence.
• For option D: When \( n = 2 \), \( a(2)=110\times2-15=220 - 15 = 205

eq80 \), so D is wrong.

Answer:

C. \( a(n)=-15n + 110 \)