Key Concepts and Formulas

• Arithmetic Progression (A.P.): A sequence where the difference between consecutive terms is constant. The $n$-th term is given by $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
• Algebraic Manipulation: Solving equations involving variables and fractions.

Step-by-Step Solution

Step 1: Understand the Given Information and Identify Unknowns We are given two arithmetic progressions, denoted by sequences $\{a_n\}$ and $\{b_n\}$. We have the following information:

• $a_1 = 2$
• $a_{10} = 3$
• $a_1 b_1 = 1$
• $a_{10} b_{10} = 1$ We need to find the value of $a_4 b_4$.

Step 2: Determine the First Term ($b_1$) and Tenth Term ($b_{10}$) of the $b$ sequence Using the given product relations and the value of $a_1$: $a_1 b_1 = 1$ $2 \cdot b_1 = 1$ $b_1 = \frac{1}{2}$

Using the given product relations and the value of $a_{10}$: $a_{10} b_{10} = 1$ $3 \cdot b_{10} = 1$ $b_{10} = \frac{1}{3}$

Step 3: Calculate the Common Difference ($d_a$) for the $a$ sequence The formula for the $n$-th term of an A.P. is $a_n = a_1 + (n-1)d$. For the $a$ sequence, we use $n=10$: $a_{10} = a_1 + (10-1)d_a$ $3 = 2 + 9d_a$ $1 = 9d_a$ $d_a = \frac{1}{9}$

Step 4: Calculate the Common Difference ($d_b$) for the $b$ sequence Similarly, for the $b$ sequence, we use $n=10$: $b_{10} = b_1 + (10-1)d_b$ $\frac{1}{3} = \frac{1}{2} + 9d_b$ To find $9d_b$, subtract $\frac{1}{2}$ from both sides: $9d_b = \frac{1}{3} - \frac{1}{2}$ Find a common denominator (6): $9d_b = \frac{2}{6} - \frac{3}{6}$ $9d_b = -\frac{1}{6}$ $d_b = -\frac{1}{54}$

Step 5: Calculate the Fourth Term ($a_4$) of the $a$ sequence Using the formula $a_n = a_1 + (n-1)d$ with $n=4$: $a_4 = a_1 + (4-1)d_a$ $a_4 = a_1 + 3d_a$ Substitute the values $a_1 = 2$ and $d_a = \frac{1}{9}$: $a_4 = 2 + 3 \left(\frac{1}{9}\right)$ $a_4 = 2 + \frac{3}{9}$ $a_4 = 2 + \frac{1}{3}$ Find a common denominator (3): $a_4 = \frac{6}{3} + \frac{1}{3}$ $a_4 = \frac{7}{3}$

Step 6: Calculate the Fourth Term ($b_4$) of the $b$ sequence Using the formula $b_n = b_1 + (n-1)d$ with $n=4$: $b_4 = b_1 + (4-1)d_b$ $b_4 = b_1 + 3d_b$ Substitute the values $b_1 = \frac{1}{2}$ and $d_b = -\frac{1}{54}$: $b_4 = \frac{1}{2} + 3 \left(-\frac{1}{54}\right)$ $b_4 = \frac{1}{2} - \frac{3}{54}$ $b_4 = \frac{1}{2} - \frac{1}{18}$ Find a common denominator (18): $b_4 = \frac{9}{18} - \frac{1}{18}$ $b_4 = \frac{8}{18}$ Simplify the fraction: $b_4 = \frac{4}{9}$

Step 7: Calculate the product $a_4 b_4$ Multiply the calculated values of $a_4$ and $b_4$: $a_4 b_4 = \left(\frac{7}{3}\right) \times \left(\frac{4}{9}\right)$ $a_4 b_4 = \frac{7 \times 4}{3 \times 9}$ $a_4 b_4 = \frac{28}{27}$

Common Mistakes & Tips

• Fraction Arithmetic Errors: Be meticulous with fraction addition, subtraction, and multiplication. Errors in finding common denominators or simplifying fractions are common.
• Misinterpreting Product Condition: Remember that $a_n b_n = 1$ is given only for $n=1$ and $n=10$, not for all $n$. Do not assume $b_n = 1/a_n$ for all terms.
• Separate Calculations: Treat the $a$ sequence and the $b$ sequence as independent A.P.s after determining their initial terms and common differences.

Summary We were given information about two arithmetic progressions, $a_n$ and $b_n$. By using the given values $a_1$, $a_{10}$, $a_1 b_1 = 1$, and $a_{10} b_{10} = 1$, we first determined the first and tenth terms of the $b$ sequence. Then, we calculated the common differences for both sequences. Finally, we found the fourth terms, $a_4$ and $b_4$, and computed their product.

The final answer is $\boxed{\frac{28}{27}}$, which corresponds to option (D).