Question

2) The number of streetlights in a town is growing linearly. Four months ago (n = 0) there were 130 lights. Now (n = 4) there are 154 lights. If this trend continues, a) Find an explicit formula for the number of lights in month n. b) How many months will it take to reach 200 lights?

Answer 1

Answer:

Step-by-step explanation:

Since the number of streetlights in a town is growing linearly, the street lights are increasing in arithmetic progression.

The formula for determining the nth term of an arithmetic sequence is expressed as

Tn = a + (n - 1)d

Where

a represents the first term of the sequence.

d represents the common difference.

n represents the number of terms in the sequence.

For n = 0, T0 = 130, therefore,

130 = a + (0 - 1)d

130 = a - d - - - - - - - - - -1

For n = 4, T4 = 154, therefore,

154 = a + (4 - 1)

154 = a + 3d - - - - - - - - - -2

Subtracting equation 2 from equation 1, it becomes

- 24 = - 4d

d = - 24/- 4 = 6

Substituting d = 6 into equation 1, it becomes

130 = a - 6

a = 130 + 6 = 136

a) The explicit formula for the number of lights in n months is

Tn = 136 + 6(n - 1)

b) The number of months that it will take to reach 200 lights is

200 = 136 + 6(n - 1)

200 - 136 = 6(n - 1)

64 = 6(n - 1)

6n - 6 = 64

6n = 64 + 6 = 70

n = 70/6 = 11.66

Approximately 12 months