Worksheet: Finding the nth Term of an Arithmetic Sequence, Set 20

Topic

Sequences and Series

Description

The final set in this comprehensive series on finding the nth term of arithmetic sequences presents the most challenging and complex problems. The theory behind finding the nth term is grounded in the principle of linear progression. The formula a_n = a + (n - 1)d represents this linear relationship, allowing us to predict any term in the sequence based on its position and the sequence's characteristics.

This concept is fundamental in mathematics and has wide-ranging applications. In finance, it can be used to calculate annuities or analyze linear depreciation. In urban planning, it can model the growth of traffic congestion over time. In data science, it forms the basis for understanding more complex sequences and series used in predictive modeling.

Teacher's Script: "Let's explore a challenging scenario. An environmental study shows that a lake's fish population is decreasing by 80 fish each year. In 2020, there were 3,200 fish, and by 2025, there were only 2,800 fish. Can we model this and predict future populations? Let's use our arithmetic sequence knowledge. We have a₁ + 5d = 2,800 and a₁ = 3,200. Solving these, we confirm d = -80. Our formula is thus P(n) = 3,200 + (n - 1)(-80) = 3,280 - 80n, where n is the number of years since 2020. Using this, we can predict the population in any future year and even determine when intervention might be necessary to preserve the ecosystem."

For a complete collection of terms related to Sequences and Series click on this link: Finding the nth Term of an Arithmetic Sequence Worksheet Collection.