The nth term of the progression $\frac{4}{2}$, $\frac{7}{3}$, $\frac{10}{4}$, $\frac{13}{5}$ is ...

Analysis

To find the $n^{th}$ term of the sequence $\frac{4}{2}, \frac{7}{3}, \frac{10}{4}, \frac{13}{5}, \dots$, we need to look at the numerator and the denominator separately to identify the pattern in each.

Concept

A sequence where the difference between consecutive terms is constant is called an Arithmetic Progression (AP). The formula for the $n^{th}$ term of an AP is: $T_n = a + (n - 1)d$ Where:

• $a$ is the first term.
• $d$ is the common difference.

Step-by-Step Solution

1. Analyze the Numerators: The numerators are: $4, 7, 10, 13, \dots$

• First term ($a$) = $4$
• Common difference ($d$) = $7 - 4 = 3$
• Using the formula: $T_n = 4 + (n - 1)3$
• $T_n = 4 + 3n - 3$
• Numerator = $3n + 1$

2. Analyze the Denominators: The denominators are: $2, 3, 4, 5, \dots$

• First term ($a$) = $2$
• Common difference ($d$) = $3 - 2 = 1$
• Using the formula: $T_n = 2 + (n - 1)1$
• $T_n = 2 + n - 1$
• Denominator = $n + 1$

3. Combine the results: By placing the numerator over the denominator, we get the $n^{th}$ term: $\frac{3n + 1}{n + 1}$

Verification

Let's test the formula with $n = 1$: $\frac{3(1) + 1}{1 + 1} = \frac{4}{2}$ (Matches the first term)

Let's test with $n = 4$: $\frac{3(4) + 1}{4 + 1} = \frac{13}{5}$ (Matches the fourth term)

The formula is correct. Comparing this to the given options, it matches option B.

Exam Tip

When you see a sequence involving fractions, always treat the numerator and denominator as two separate sequences. Don't try to find a single formula for the whole fraction at once; it is much easier to solve them individually and then combine them!