Problem 30
Question
The first term \(a_{1}\) and the common difference d of an arithmetic sequence are given. Find the fifth term and the formula for the nth term. $$a_{1}=\pi, d=\frac{1}{5}$$
Step-by-Step Solution
Verified Answer
2) What is the formula for the nth term of the arithmetic sequence?
Answers:
1) The fifth term of the arithmetic sequence is \(a_5 = \pi + \frac{4}{5}\).
2) The formula for the nth term of the arithmetic sequence is \(a_n = \pi + (n-1)\left(\frac{1}{5}\right)\).
1Step 1: Insert the given values into the formula for the fifth term
To find the fifth term, we need to substitute the given values \(a_1 = \pi\), \(d = \frac{1}{5}\), and \(n = 5\) into the arithmetic sequence formula:
$$a_5 = \pi + (5-1)\left(\frac{1}{5}\right)$$
2Step 2: Calculate the fifth term
Now we will simplify and calculate the expression for \(a_5\):
$$a_5 = \pi + (4)\left(\frac{1}{5}\right) = \pi + \frac{4}{5}$$
Therefore, the fifth term of the arithmetic sequence is \(a_5 = \pi + \frac{4}{5}\).
3Step 3: Write a general formula for the nth term
We can write a formula for the nth term of the arithmetic sequence by substituting the given values \(a_1 = \pi\) and \(d = \frac{1}{5}\) into the arithmetic sequence formula:
$$a_n = \pi + (n-1)\left(\frac{1}{5}\right)$$
This is the desired formula for the nth term of the arithmetic sequence.
Key Concepts
Arithmetic Sequence FormulaArithmetic Sequence TermsMathematical Induction
Arithmetic Sequence Formula
Understanding the arithmetic sequence formula is fundamental to dealing with such progressions in mathematics. An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This difference is often referred to as the 'common difference', denoted as 'd'. The formula to find the nth term of an arithmetic sequence is given by:
Here, \(a_n\) is the nth term of the sequence, \(a_1\) is the first term, and 'n' is the term number. Using this formula, we can determine any term's value within the sequence provided we know the first term and the common difference.
For example, given the first term \(a_1 = \) and the common difference \(d = \frac{1}{5}\), if we wanted to find the 5th term, we would substitute these values into the formula to get:
After simplifying, we would find that \(a_5 = \ + \frac{4}{5}\). This formula's simplicity allows for quick calculations and is essential for solving problems that entail arithmetic sequences.
- \(a_n = a_1 + (n - 1)d\)
Here, \(a_n\) is the nth term of the sequence, \(a_1\) is the first term, and 'n' is the term number. Using this formula, we can determine any term's value within the sequence provided we know the first term and the common difference.
For example, given the first term \(a_1 = \) and the common difference \(d = \frac{1}{5}\), if we wanted to find the 5th term, we would substitute these values into the formula to get:
- \(a_5 = \ + (5 - 1)\left(\frac{1}{5}\right)\)
After simplifying, we would find that \(a_5 = \ + \frac{4}{5}\). This formula's simplicity allows for quick calculations and is essential for solving problems that entail arithmetic sequences.
Arithmetic Sequence Terms
An arithmetic sequence is defined by its terms, which are the elements of the sequence. Each term in the sequence has a specific position, often denoted by an index that starts at 1 for the first term. The terms in an arithmetic sequence are usually denoted as \(a_1\), \(a_2\), \(a_3\), and so forth. The key characteristic that these terms share is that they follow a specific pattern determined by the common difference 'd'.
The first term \(a_1\) sets the starting point, and each subsequent term is found by adding the common difference to the previous term. In other words, the second term is \(a_2 = a_1 + d\), the third term is \(a_3 = a_2 + d\), and so on. By understanding the relationship between the terms, we can construct the entire sequence, predict future terms, or find missing values within the sequence.
The first term \(a_1\) sets the starting point, and each subsequent term is found by adding the common difference to the previous term. In other words, the second term is \(a_2 = a_1 + d\), the third term is \(a_3 = a_2 + d\), and so on. By understanding the relationship between the terms, we can construct the entire sequence, predict future terms, or find missing values within the sequence.
Mathematical Induction
Mathematical induction is a powerful proof technique used in mathematics to establish the truth of infinitely many cases by proving two essential steps: the base case and the inductive step. While it is not explicitly needed to calculate terms in an arithmetic sequence, knowing how to prove properties about sequences using induction can be quite beneficial.
For the base case, one shows that a property holds for the first item in the sequence, often \(n = 1\). Then, assuming the property holds for an arbitrary natural number \(k\), the inductive step involves proving that the property also holds for \(k + 1\). If both of these steps are satisfied, we can conclude that the property holds for all natural numbers in the sequence.
This method is not just limited to arithmetic sequences but is also widely applicable across various mathematical concepts. It helps in understanding and verifying the underlying principles that govern the behavior of numbers and sets within a particular context.
For the base case, one shows that a property holds for the first item in the sequence, often \(n = 1\). Then, assuming the property holds for an arbitrary natural number \(k\), the inductive step involves proving that the property also holds for \(k + 1\). If both of these steps are satisfied, we can conclude that the property holds for all natural numbers in the sequence.
This method is not just limited to arithmetic sequences but is also widely applicable across various mathematical concepts. It helps in understanding and verifying the underlying principles that govern the behavior of numbers and sets within a particular context.
Other exercises in this chapter
Problem 30
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