Question

The ratio of the sum of \(n\) terms of two arithmetic progressions is given by \((2 n+3):(5 n-7)\). Find the ratio of their \(\mathrm{nth}\) terms. (1) \((4 n+5):(10 n+2)\) (2) \((4 n+1):(10 n-12)\) (3) \((4 n-1):(10 n+8)\) (4) \((4 n-5):(10 n-2)\)

Short answer

The ratio of the nth terms of the two arithmetic progressions is \((4n-5):(10n-2)\).

Step-by-step solution

Write the general formula for the sum of an arithmetic progression

The formula for the sum of the first \(n\) terms of an arithmetic progression with a first term \(a\) and a common difference \(d\) is: \(S_n = n \cdot \frac{2a + (n - 1) \cdot d}{2}\)

Define the two arithmetic progression sums

Let the first arithmetic progression have the first term \(a_1\) and the common difference \(d_1\). Define the sum of its \(n\) terms as \(S_{1n}\). Similarly, let the second arithmetic progression have the first term \(a_2\) and the common difference \(d_2\). Define the sum of its \(n\) terms as \(S_{2n}\).

Write the given ratio in equation form

We are given that the ratio of the sum of \(n\) terms of the two arithmetic progressions is \((2n+3):(5n-7)\). We can write this as an equation: \(\frac{S_{1n}}{S_{2n}} = \frac{(2n+3)}{(5n-7)}\)

Insert the general formulas for the arithmetic progression sums

Replace \(S_{1n}\) and \(S_{2n}\) with their respective arithmetic progression sum formulas: \(\frac{n \cdot \frac{2a_1 + (n - 1) \cdot d_1}{2}}{n \cdot \frac{2a_2 + (n - 1) \cdot d_2}{2}} = \frac{(2n+3)}{(5n-7)}\)

Simplify the equation

Since both the numerator and the denominator have the term \(n \cdot \frac{1}{2}\), we can cancel these terms out. The resulting equation is: \(\frac{2a_1 + (n - 1) \cdot d_1}{2a_2 + (n - 1) \cdot d_2} = \frac{(2n+3)}{(5n-7)}\)

Set the equation equal to the nth term ratio

We want to find the ratio of the nth terms of the two arithmetic progressions. Recognizing that the general formula for the nth term of an arithmetic progression is \(a_n = a + (n-1) \cdot d\), we can substitute these into the equation and find the ratio of the nth terms: \(\frac{a_1 + (n-1) \cdot d_1}{a_2 + (n-1) \cdot d_2} = \frac{(2n+3)}{(5n-7)}\)

Compare the simplified equation with the answer choices

In the form we have the equation, it directly shows the nth term ratios. Now, compare this to the given answer choices. \((4n+5):(10n+2)\) is not equal to \(\frac{(2n+3)}{(5n-7)}\) \((4n+1):(10n-12)\) is not equal to \(\frac{(2n+3)}{(5n-7)}\) \((4n-1):(10n+8)\) is not equal to \(\frac{(2n+3)}{(5n-7)}\) \((4n-5):(10n-2)\) is equal to \(\frac{(2n+3)}{(5n-7)}\) Hence, the ratio of the nth terms of the two arithmetic progressions is \((4n-5):(10n-2)\). So, the correct answer is option (4).

Key concepts

Sum of Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. Calculating the sum of an arithmetic progression allows us to determine the total of all terms up to a particular number of terms,usually denoted by \( n \). The formula for the sum of the first \( n \) terms of an AP, where \( a \) is the first term and \( d \) is the common difference, is:

\[ S_n = n \cdot \frac{2a + (n - 1) \cdot d}{2} \]
This formula gives a straightforward method to find out how much the first \( n \) terms of the sequence add up to. In problems, this formula helps us compare different series, especially when given their sum in a certain ratio.

• \( n \) stands for the number of terms.
• \( a \) represents the first term of the sequence.
• \( d \) is the common difference.

Knowing the sum of an arithmetic progression helps us solve more complex problems related to sequences and series.

nth term of a sequence

Finding the \( n \)-th term of an arithmetic progression allows us to identify any specific term in the sequence without having to list all preceding ones. This is crucial in understanding or predicting the behavior of a sequence directly. The formula for the \( n \)-th term is:

\[ a_n = a + (n-1) \cdot d \]
This sequence formula helps find any term's exact value by utilizing the first term and the common difference.

• This method allows efficient computation, especially when dealing with long sequences.
• The formula reveals the linear nature of arithmetic sequences, where terms scale uniformly based on their position \( n \).

By mastering this formula, we can easily calculate or manipulate specific terms in an AP, enabling us to solve problems concerning sequence ratios or sums.

Ratio of Sequences

The ratio of sequences refers to comparing the terms in two sequences or the sum of terms in two sequences. This is particularly important in problems where different sequences are compared based on certain properties.
In arithmetic progressions, the ratio can be applied both to specific terms and sums of terms.
Often, you'll encounter problems that give a ratio like \((2n+3):(5n-7)\), which can illustrate relationships between two different sequence sums or terms.

• Ratios provide insight into how the sequences grow relative to each other.
• They can show equivalence or particular proportional relationships between sums or terms.
• This can be applied in real-world situations where sequences describe quantities such as time intervals or distances.

Understanding ratios is crucial for analyzing multiple sequences and drawing comparisons effectively.

Common Difference

The common difference in an arithmetic progression is the constant amount by which each term increases (or decreases) when moving from one term to the next.
The common difference plays a critical role in defining the nature of the sequence and is a core part of both the sum and nth term formulas.

• It is denoted by \( d \).
• A positive common difference results in an increasing sequence.
• A negative common difference leads to a decreasing sequence.
With a known common difference, one can reconstruct the entire sequence given just the first term. This property is key in both analyzing and predicting patterns in number sequences.