The 10th term of an AP is 32. If the first term is 3/2, what is the 4th term?

Analysis

To solve this problem, we need to use the formula for the $n^{th}$ term of an Arithmetic Progression (AP): $T_n = a + (n - 1)d$

Where:

• $T_n$ is the $n^{th}$ term.
• $a$ is the first term.
• $n$ is the position of the term.
• $d$ is the common difference.

Concept

We are given:

• The 10th term ($T_{10}$) = $32$
• The first term ($a$) = $\frac{3}{2}$
• We need to find the 4th term ($T_4$).

First, we must find the common difference ($d$) using the information for the 10th term, then use that $d$ to calculate the 4th term.

Step-by-Step Solution

Step 1: Find the common difference ($d$) Using $T_{10} = a + (10 - 1)d$: $32 = \frac{3}{2} + 9d$

Subtract $\frac{3}{2}$ from both sides: $32 - \frac{3}{2} = 9d$ $\frac{64 - 3}{2} = 9d$ $\frac{61}{2} = 9d$

Divide both sides by 9: $d = \frac{61}{2 \times 9} = \frac{61}{18}$

Step 2: Find the 4th term ($T_4$) Using $T_4 = a + (4 - 1)d$: $T_4 = \frac{3}{2} + 3d$

Substitute $d = \frac{61}{18}$: $T_4 = \frac{3}{2} + 3(\frac{61}{18})$ $T_4 = \frac{3}{2} + \frac{61}{6}$

Find a common denominator (which is 6): $T_4 = \frac{9}{6} + \frac{61}{6}$ $T_4 = \frac{70}{6}$

Simplify the fraction by dividing both numerator and denominator by 2: $T_4 = \frac{35}{3}$

Conclusion

The calculated 4th term is $\frac{35}{3}$, which matches option A.

Exam Tip: When dealing with fractions in an AP, always convert the whole numbers to fractions with the same denominator as early as possible. This prevents calculation errors when subtracting or adding terms.