Question

Find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=\sqrt{2} ; \quad d=\sqrt{2} $$

Short answer

a_{51} = 51 sqrt{2}

Step-by-step solution

Identify the general formula for the nth term

The nth term of an arithmetic sequence can be found using the formula: a_{n} = a_{1} + (n-1)d

Substitute the given values

We are given the first term ( a_{1} ) as a_{1} = sqrt{2} and the common difference ( d ) as d = sqrt{2} . Substitute these into the general formula: a_{n} = sqrt{2} + (n-1) sqrt{2}

Simplify the formula

Simplify the expression: a_{n} = sqrt{2} + (n-1) sqrt{2} = sqrt{2} + n sqrt{2} - sqrt{2} = n sqrt{2}

Find the 51st term

Substitute n = 51 into the simplified formula: a_{51} = 51 sqrt{2}

Key concepts

nth term formula

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. To find the nth term in an arithmetic sequence, we use the nth term formula:

\(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_n\) represents the nth term, \(a_1\) is the first term, and \(d\) is the common difference between terms. This formula helps us calculate any term in the sequence if we know the first term and the common difference. In our given problem, we need this formula to find the 51st term.

first term

The first term of an arithmetic sequence is the starting point of the sequence and is denoted as \(a_1\). In the given exercise, the first term \(a_1\) is \sqrt{2}\.

Understanding the first term is crucial because it sets the base for the entire sequence. Think of it as the initial step that influences all further terms. If you change the first term, the entire sequence shifts even if the common difference remains the same.

common difference

The common difference in an arithmetic sequence is the difference between any two successive terms and is denoted as \(d\). It is crucial because it dictates the increment between the terms.

For our problem, \(d = \sqrt{2}\). This means every subsequent term is \sqrt{2}\ units larger than the previous one. In simpler terms, if you start at \sqrt{2}\, the next term will be \2\sqrt{2}\, and so on. Think of the common difference as the step size you keep adding to move from one term to the next.

simplifying expressions

Simplifying expressions is the process of making a mathematical expression as simple as possible. It often involves combining like terms and making calculations more straightforward.

In the exercise, we simplified the expression to find the nth term:
This was done as follows:

1. Start with the nth term formula: \(a_n = a_1 + (n-1) \cdot d\)
2. Substitute \sqrt{2}\ for \(a_1\) and \sqrt{2}\ for \(d\): \(a_n = \sqrt{2} + (n-1) \sqrt{2}\)
3. Distribute \(\sqrt{2}\) within the parentheses: \(a_n = \sqrt{2} + n \sqrt{2} - \sqrt{2}\)
4. Combine like terms to get the simplified result: \(a_n = n \sqrt{2}\).

Simplifying expressions is a critical skill in math. It makes equations easier to work with and helps us understand the underlying pattern of a sequence.

finding specific terms

After simplifying the nth term formula, we can use it to find any specific term in the sequence. In this exercise, we wanted to find the 51st term.
Using our simplified formula \(a_n = n \sqrt{2}\):

1. Substitute \51\ for \(n\).
2. This gives \(a_{51} = 51 \sqrt{2}\).

You can use this process to find any term in an arithmetic sequence by simply plugging in the desired term number into the simplified nth term formula. It's a straightforward way to get specific terms without recomputing the entire sequence every time.