Problem 3
Question
Find the first four terms and the 100 th term of the sequence. $$a_{n}=\frac{1}{n+1}$$
Step-by-Step Solution
Verified Answer
The first four terms are \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \). The 100th term is \( \frac{1}{101} \).
1Step 1: Finding the first term
To find the first term of the sequence, substitute \( n = 1 \) into the given formula. \[ a_1 = \frac{1}{1+1} = \frac{1}{2} \] So, the first term is \( \frac{1}{2} \).
2Step 2: Finding the second term
Substitute \( n = 2 \) into the formula to find the second term. \[ a_2 = \frac{1}{2+1} = \frac{1}{3} \] Thus, the second term is \( \frac{1}{3} \).
3Step 3: Finding the third term
Substitute \( n = 3 \) into the formula to find the third term. \[ a_3 = \frac{1}{3+1} = \frac{1}{4} \] Therefore, the third term is \( \frac{1}{4} \).
4Step 4: Finding the fourth term
To find the fourth term, substitute \( n = 4 \) into the formula. \[ a_4 = \frac{1}{4+1} = \frac{1}{5} \] So, the fourth term is \( \frac{1}{5} \).
5Step 5: Finding the 100th term
Substitute \( n = 100 \) into the formula to find the 100th term. \[ a_{100} = \frac{1}{100+1} = \frac{1}{101} \] Thus, the 100th term is \( \frac{1}{101} \).
Key Concepts
Sequences in MathematicsCalculating Terms in a SequenceUtilizing Mathematical Formulas
Sequences in Mathematics
A sequence in mathematics is a set of numbers that follow a specific pattern. Sequences play a crucial role in various mathematical concepts and problems.
They help in understanding patterns, trends, and structures of numbers. There are different types of sequences, such as arithmetic, geometric, and harmonic, among others.
What differentiates one type from another is the rule they follow to progress from one term to the next. For example, in an arithmetic sequence, each number is obtained by adding a fixed value to the previous number. Understanding sequences allows mathematicians and students alike to grasp recurring structures in numbers, which is useful in predictions and in proving theories. In this exercise, understanding the concept of sequences helps in comprehending how each subsequent term is calculated based on the formula given.
They help in understanding patterns, trends, and structures of numbers. There are different types of sequences, such as arithmetic, geometric, and harmonic, among others.
What differentiates one type from another is the rule they follow to progress from one term to the next. For example, in an arithmetic sequence, each number is obtained by adding a fixed value to the previous number. Understanding sequences allows mathematicians and students alike to grasp recurring structures in numbers, which is useful in predictions and in proving theories. In this exercise, understanding the concept of sequences helps in comprehending how each subsequent term is calculated based on the formula given.
Calculating Terms in a Sequence
Finding a specific term in a sequence can be straightforward if you have the formula that defines it. Term calculation involves substituting the term number into the sequence's formula. This gives the specific value for that position in the sequence.
Let's look at the sequence defined by the formula \( a_{n} = \frac{1}{n+1} \). To find any term, you simply replace \( n \) with the position of the term.
For example:
Let's look at the sequence defined by the formula \( a_{n} = \frac{1}{n+1} \). To find any term, you simply replace \( n \) with the position of the term.
For example:
- For the first term (\( n = 1 \)), substitute 1 into the formula to get \( a_1 = \frac{1}{1+1} = \frac{1}{2} \).
- For the second term (\( n = 2 \)), substitute 2 into the formula to find \( a_2 = \frac{1}{2+1} = \frac{1}{3} \).
- This pattern continues, allowing you to find any term in the sequence by simply adjusting \( n \).
Utilizing Mathematical Formulas
Mathematical formulas are powerful tools that allow for the efficient calculation of terms and solving of problems. Utilizing these formulas correctly is essential in getting accurate results.
In the context of sequences, a formula gives a shortcut to quickly determine the value of any term without listing all the previous terms.
Using the sequence formula \( a_n = \frac{1}{n+1} \), students can calculate complex terms, like the 100th term, without computing every prior term:
In the context of sequences, a formula gives a shortcut to quickly determine the value of any term without listing all the previous terms.
Using the sequence formula \( a_n = \frac{1}{n+1} \), students can calculate complex terms, like the 100th term, without computing every prior term:
- Plug \( n = 100 \) into the formula to find \( a_{100} = \frac{1}{100+1} = \frac{1}{101} \).
Other exercises in this chapter
Problem 3
Use Pascal's triangle to expand the expression. $$\left(x+\frac{1}{x}\right)^{4}$$
View solution Problem 3
The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (
View solution Problem 4
A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$a_{n}=\fra
View solution Problem 4
Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$
View solution