Problem 8 - Basic Engineering Mathematics

Question

Find the \(n^{\prime}\) th term of the series: \(1,4,9\), \(16,25, \ldots\)

Step-by-Step Solution

Answer

The nth term of the series is \(n^2\).

Step 1: Identify the pattern

The given series is: 1, 4, 9, 16, 25, .... Let's determine the pattern by looking at the terms. Notice that each term is a perfect square: \(1 = 1^2\), \(4 = 2^2\), \(9 = 3^2\), \(16 = 4^2\), \(25 = 5^2\). This indicates a pattern where each term is the square of a natural number.

Step 2: Express the general term using the identified pattern

Since the terms are perfect squares of consecutive natural numbers, the \(n^{th}\) term of the series can be represented as \(a_n = n^2\), where \(n\) is 1, 2, 3, and so on.

Step 3: Write the formula for the nth term

Based on the identified pattern and the way we have expressed the general term, the formula for the \(n^{th}\) term in the series is given by \(a_n = n^2\).

Key Concepts

Perfect Squares

In mathematics, a perfect square is a number that can be expressed as the square of an integer. For example:

• The number 1 is a perfect square because it can be written as \(1^2\)
• The number 4 is a perfect square because it is \(2^2\)
• The number 9 is \(3^2\)
• Continuing on, 16 is \(4^2\) and 25 is \(5^2\).

These numbers are called perfect squares because they are the product of an integer multiplied by itself.
They have a special significance in mathematics, particularly in number theory and algebra. Recognizing perfect squares helps us find patterns in sequences and solve equations more efficiently.

nth Term Formula

The nth term formula of a sequence provides a way to find any term in the sequence without listing all previous terms. It is particularly useful for long sequences.
For the exercise series: 1, 4, 9, 16, 25, ..., each term is a square of its position in the sequence.
To represent this pattern algebraically, we use the nth term formula \( a_n = n^2 \).

This formula tells us that:

• To find the first term (\( n = 1 \)), calculate \( a_1 = 1^2 = 1 \)
• For the second term (\( n = 2 \)), it's \( a_2 = 2^2 = 4 \)
• The third term (\( n = 3 \)) is \( a_3 = 3^2 = 9 \)
• This pattern continues indefinitely, following the rule \( a_n = n^2 \)

By applying the nth term formula, you can determine any term in the sequence without writing each out step by step. This makes it convenient for predicting or calculating future terms, and serves as an essential tool in understanding number sequences.

Natural Numbers

Natural numbers are the set of positive integers beginning from 1, extending indefinitely upwards. They include:

• 1, 2, 3, 4, 5, ...

Natural numbers are often used to describe order or position in a sequence.
In our sequence, each perfect square corresponds to a natural number squared:

• 1 is the first natural number and its square is 1 (\(1^2 = 1\))
• 2 is the second natural number and its square is 4 (\(2^2 = 4\))
• 3 is the third natural number and its square is 9 (\(3^2 = 9\))
• And so on.

By using natural numbers as exponents in the context of this sequence, we can systematically understand and calculate the values of all the terms easily. The relationship of natural numbers with perfect squares shows the structural significance they have in mathematical sequences.