The city of Lightsville currently has 137 streetlights. As...

The city of Lightsville currently has 137 streetlights. As part of an urban renewal program, the city council has decided to install and have operational 2 additional streetlights at the end of each week for the next 52 weeks. Each streetlight costs $\$ 1$ to operate for 1 week. (a) How many streetlights will the city have at the end of 38 weeks? (b) How many streetlights will the city have at the end of $N$ weeks? (Assume $N \leq 52 .)$ (c) What is the cost of operating the original 137 lights for 52 weeks? (d) What is the additional cost for operating the newly installed lights for the 52 -week period during which they are being installed?

The city of Lightsville currently has 137 streetlights. As part of an urban renewal program, the city council has decided to install and have operational 2 additional streetlights at the end of each week for the next 52 weeks. Each streetlight costs $\$ 1$ to operate for 1 week.
(a) How many streetlights will the city have at the end of 38 weeks?
(b) How many streetlights will the city have at the end of $N$ weeks? (Assume $N \leq 52 .)$
(c) What is the cost of operating the original 137 lights for 52 weeks?
(d) What is the additional cost for operating the newly installed lights for the 52 -week period during which they are being installed?

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a fixed constant (called the common difference) to the previous term. This concept is used to model situations where a quantity increases or decreases by the same amount at regular intervals, such as adding a fixed number of streetlights each week.

General Term Formula for Arithmetic Sequences

The general term of an arithmetic sequence can be calculated using the formula: a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula is crucial for determining the value of a quantity at any given point, such as the number of streetlights after a specific number of weeks.

Arithmetic Series Summation

An arithmetic series is the sum of the terms of an arithmetic sequence. It involves adding together quantities that increase by a constant amount and is commonly calculated using the formula: S_n = n/2 × (first term + last term) or S_n = n/2 × [2a_1 + (n - 1)d]. This concept helps in determining total quantities or costs that accumulate over time, such as the cost of operating newly added streetlights over several weeks.

Unit Cost Calculation

Unit cost calculation involves multiplying the number of units by a fixed cost per unit to determine an overall cost. In practical applications, this means calculating the cost to operate equipment or perform tasks over time by considering how many units are being used and how long they are used. This technique is applied to determine total operating costs when each streetlight has an associated weekly operational expense.

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