Question

Deduce a recursion relation for each series. Use it to predict the next two terms. $$5+15+45+\cdots$$

Short answer

The recursion relation is T_{n+1} = 3 * T_n. The next two terms of the series are 135 and 405.

Step-by-step solution

Identify the Pattern

Analyze the given series to determine the relationship between consecutive terms. For the series 5, 15, 45, we notice that every term after the first can be generated by multiplying the previous term by 3.

Write the Recursion Formula

Express the pattern discovered into a recursive formula. For the series given, the recursive formula can be written as: for every term after the first, T_{n+1} = 3 * T_n, where T_n represents the n-th term.

Apply Recursion to Predict the Next Terms

Use the recursive formula to find the next two terms of the series. Multiply the last given term by 3 to find the subsequent term. For the next term after 45, we calculate T_4 = 3 * 45 = 135. Then for the term after that, T_5 = 3 * 135 = 405.

Key concepts

Series and Sequences

When we talk about mathematics, series and sequences are fundamental concepts that often come up in various contexts. A sequence is a list of numbers arranged in a particular order that often follow a specific pattern. A series, on the other hand, is the sum of the terms of a sequence.

Sequences can be finite, with a limited number of terms, or infinite, continuing perpetually according to their defined rule. They are categorized based on their pattern of progression—some increase (or decrease) by a fixed amount, called an arithmetic sequence, while others may multiply (or divide) by a consistent rate, known as a geometric sequence.

Understanding the underlying rule behind a sequence is key to exploring its properties and predicting future terms. For students tackling problems involving sequences, grasping this pattern is the pivotal step in moving from confusion to clarity.

Geometric Progression

A geometric progression (GP) stands out as a type of sequence where each term after the first is found by multiplying the previous one by a constant called the common ratio. This ratio remains the same throughout the progression.

For instance, the given series in the exercise, 5, 15, 45, ... is part of a geometric progression where each term is three times its predecessor. This common ratio of 3 acts as the thread weaving through the series, helping us understand and predict subsequent terms. GP is significant in various disciplines, including finance for calculating compound interest, and in the physical sciences for understanding phenomena that grow or decay exponentially.

By recognizing a series as a geometric progression, students can unlock a powerful tool for solving complex problems with a simple recursive formula.

Recursive Formula

A recursive formula allows us to determine the subsequent terms in a sequence based on its preceding terms. It's essentially a mathematical recipe that gives instructions on how to progress from one term to the next. In the context of geometric progressions, the recursion relation comprises of a starting term (also called the first term) and a method for finding each term after the first, by multiplying the previous term by the common ratio of the sequence.

The formula provided in the exercise solution, \( T_{n+1} = 3 \cdot T_n \) is a classic example of a recursion relation in a geometric progression. Students using recursive formulas benefit from understanding how a sequence evolves and grows—this grasp of recursive processes is not only relevant in pure mathematics, but also in computational algorithms and data structures.