Building a tunnel — first scenario
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.


a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?

Verified step by step guidance

1. Identify the type of sequence described in the problem. Since each week the crew digs 0.95 times the distance dug the previous week, this is a geometric sequence where each term is multiplied by a common ratio \(r = 0.95\).
2. Write the general formula for the distance dug in the \(n^{th}\) week. The first term \(a_1\) is 100 meters, so the distance dug in week \(n\) is given by \(a_n = a_1 \times r^{n-1} = 100 \times 0.95^{n-1}\).
3. To find the total distance dug after \(N\) weeks, recognize that this is the sum of the first \(N\) terms of a geometric sequence. Use the formula for the sum of the first \(N\) terms: \(S_N = a_1 \times \frac{1 - r^N}{1 - r}\).
4. Apply the sum formula to calculate the total distance dug after 10 weeks by substituting \(N=10\), \(a_1=100\), and \(r=0.95\) into the formula: \(S_{10} = 100 \times \frac{1 - 0.95^{10}}{1 - 0.95}\).
5. Similarly, calculate the total distance dug after 20 weeks by substituting \(N=20\) into the sum formula: \(S_{20} = 100 \times \frac{1 - 0.95^{20}}{1 - 0.95}\). For a general number of weeks \(N\), use \(S_N = 100 \times \frac{1 - 0.95^{N}}{1 - 0.95}\).

Key Concepts

Geometric Sequences

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant ratio. In this problem, the distance dug each week decreases by a factor of 0.95, making it a geometric sequence with first term 100 and common ratio 0.95.

Sum of a Finite Geometric Series

The total distance dug over multiple weeks is the sum of the terms in the geometric sequence. The sum of the first n terms of a geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

Generalization to N Terms

To find the distance dug after N weeks, the formula for the sum of the first N terms of the geometric series is used. This allows calculation of the total distance for any number of weeks, not just specific values like 10 or 20.