Sequences and Series: Understanding Different Patterns

What are the first ten terms of the following sequences?

1. The nth term is √n

2. The first two terms in the sequence are 1. The rest of the terms are the sum of the two preceding terms.

3. The nth term is the largest integer k such that k! ≤ n.

Answer:

1. The first ten terms of the sequence defined by the nth term as √n are: 1, √2, √3, √4, √5, √6, √7, √8, √9, √10.

2. The sequence with the first two terms being 1 and the rest of the terms being the sum of the two preceding terms is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

3. The first ten terms of the sequence defined by the nth term as the largest integer k such that k! ≤ n are: 1, 2, 2, 3, 4, 5, 5, 6, 7, 7.

Sequences and series are an essential part of mathematics. They follow different patterns and rules that help us understand the relationships between numbers. In the given data, we have three sequences with unique definitions for their nth terms.

Sequence 1: √n

This sequence is generated by taking the square root of each positive integer starting from 1. The first term is 1, and the subsequent terms are the square roots of the integers incrementing by 1. For example, the second term is √2, the third term is √3, and so on up to the tenth term √10.

Sequence 2: Fibonacci Sequence

The Fibonacci sequence starts with two initial terms, 1 and 1. To find the next term, you sum the two preceding terms. This pattern continues, creating a sequence of numbers where each term is the sum of the two before it. The first ten terms in this sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55.

Sequence 3: Largest Integer k such that k! ≤ n

In this sequence, we find the largest integer, k, such that k! (k factorial) is less than or equal to the given value of n. For instance, when n = 1, the largest integer k for which k! ≤ 1 is 1. The sequence continues with increasing values of k as long as k! remains less than or equal to n.

Understanding these different sequences and their patterns can be helpful in various mathematical applications. By recognizing these sequences, you can derive insights and solutions to a wide range of problems.