Finding the Value of n When Nth Terms of Two Arithmetic Series are Equal

The problem asks us to find the value of 'n' where the nth term of the first arithmetic series is equal to the nth term of the second arithmetic series.

We are given two series:

1. First Series: 3, 10, 17, ...
2. Second Series: 63, 65, 67, ...

Both are arithmetic series (AP) because the difference between consecutive terms is constant.

Analysing the First Arithmetic Series

The first series is 3, 10, 17, ...

• The first term, \(a_1 = 3\).
• The common difference, \(d_1\), is the difference between any term and its preceding term. \(d_1 = 10 - 3 = 7\).

The formula for the nth term of an arithmetic series is \(a_n = a_1 + (n-1)d\).

For the first series, the nth term \(a_{n1}\) is:

\(a_{n1} = 3 + (n-1)7\)

\(a_{n1} = 3 + 7n - 7\)

\(a_{n1} = 7n - 4\)

Analysing the Second Arithmetic Series

The second series is 63, 65, 67, ...

• The first term, \(a_2 = 63\).
• The common difference, \(d_2\), is the difference between any term and its preceding term. \(d_2 = 65 - 63 = 2\).

Using the formula for the nth term of an arithmetic series, \(a_n = a_1 + (n-1)d\).

For the second series, the nth term \(a_{n2}\) is:

\(a_{n2} = 63 + (n-1)2\)

\(a_{n2} = 63 + 2n - 2\)

\(a_{n2} = 2n + 61\)

Equating the Nth Terms to Find n

We are given that the nth terms of the two series are equal. Therefore, we set \(a_{n1} = a_{n2}\).

\(7n - 4 = 2n + 61\)

Now, we need to solve this linear equation for \(n\).

1. Subtract \(2n\) from both sides of the equation:

   \(7n - 2n - 4 = 2n - 2n + 61\)

   \(5n - 4 = 61\)

2. Add 4 to both sides of the equation:

   \(5n - 4 + 4 = 61 + 4\)

   \(5n = 65\)

3. Divide both sides by 5:

   \(n = \frac{65}{5}\)

   \(n = 13\)

So, the value of \(n\) for which the nth terms of the two series are equal is 13.

Let's check our answer:

• 13th term of the first series: \(a_{13,1} = 7(13) - 4 = 91 - 4 = 87\)
• 13th term of the second series: \(a_{13,2} = 2(13) + 61 = 26 + 61 = 87\)

The 13th terms are indeed equal (87).

Conclusion on Finding n

By finding the general formula for the nth term of each arithmetic series and setting them equal, we solved for \(n\). The value of \(n\) is 13.

Revision Table: Arithmetic Series Key Concepts

Additional Information: Solving Series Problems

Understanding the properties of sequences and series is fundamental in mathematics. For arithmetic series, the nth term formula is a powerful tool that allows us to find any term without listing all the terms up to that position. When dealing with problems involving two or more series, equating their terms (like the nth term) often leads to an equation that can be solved to find unknown values like 'n' or other parameters of the series.

In this specific problem, equating the nth terms resulted in a simple linear equation. Solving linear equations is a basic algebraic skill involving isolating the variable (in this case, \(n\)) by performing inverse operations on both sides of the equation.

Remember that 'n' in the context of series typically represents the term number, which must be a positive integer (\(n \ge 1\)). If solving an equation for 'n' in such a problem yields a non-integer or a non-positive value, it might indicate that no such term exists or there was an error in the setup.