Key Concepts and Formulas

• Arithmetico-Geometric Progression (AGP): A sequence where each term is the product of the corresponding terms of an Arithmetic Progression (AP) and a Geometric Progression (GP). The general form is $a, (a+d)r, (a+2d)r^2, \dots$.
• Sum of an AGP: The sum of the first $n$ terms of an AGP with first term $a$, common difference $d$, and common ratio $r$ is given by $S_n = \frac{a - (a+(n-1)d)r^n}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2}$ (for $r \neq 1$). However, for a finite number of terms, direct summation is often more practical.
• Symmetric Representation: For an odd number of terms in an AGP, representing the AP terms symmetrically around a central term ($A$) and the GP terms symmetrically around a central term ($G$) can simplify calculations.

Step-by-Step Solution

Step 1: Represent the terms of the AGP. Let the 5 terms of the AGP be $T_1, T_2, T_3, T_4, T_5$. We are given $T_3 = 2$. Let the corresponding AP have first term $A'$ and common difference $D'$. Let the corresponding GP have first term $G'$ and common ratio $R=2$. A convenient way to represent the terms of an AGP, especially when a middle term is known, is to use a symmetric form. Let the AP terms be $A-2d, A-d, A, A+d, A+2d$ and the GP terms be $g/r^2, g/r, g, gr, gr^2$. The AGP terms are then: $T_1 = (A-2d) \cdot (g/r^2)$ $T_2 = (A-d) \cdot (g/r)$ $T_3 = A \cdot g$ $T_4 = (A+d) \cdot (gr)$ $T_5 = (A+2d) \cdot (gr^2)$

We are given that the common ratio of the corresponding GP is $R=2$. We can choose the central GP term $g$ to be 1 without loss of generality. This simplifies the GP terms to $1/4, 1/2, 1, 2, 4$. Let $A$ be the central term of the AP. So, the AGP terms are: $T_1 = (A-2d) \cdot \frac{1}{4}$ $T_2 = (A-d) \cdot \frac{1}{2}$ $T_3 = A \cdot 1$ $T_4 = (A+d) \cdot 2$ $T_5 = (A+2d) \cdot 4$

Step 2: Determine the central AP term ($A$) using the given $T_3$. We are given that $T_3 = 2$. From our representation, $T_3 = A \cdot 1$. Therefore, $A = 2$.

Step 3: Set up the equation for the sum of the AGP terms. The sum of all 5 terms of the AGP is given as $\frac{49}{2}$. Substituting $A=2$ into the expressions for the AGP terms: $T_1 = (2-2d) \cdot \frac{1}{4} = \frac{2-2d}{4}$ $T_2 = (2-d) \cdot \frac{1}{2} = \frac{2-d}{2}$ $T_3 = 2$ $T_4 = (2+d) \cdot 2 = 4+2d$ $T_5 = (2+2d) \cdot 4 = 8+8d$

The sum $S_5 = T_1 + T_2 + T_3 + T_4 + T_5 = \frac{49}{2}$. $\frac{2-2d}{4} + \frac{2-d}{2} + 2 + (4+2d) + (8+8d) = \frac{49}{2}$

Step 4: Solve for the common difference ($d$) of the AP. To clear the denominators, multiply the entire equation by 4: $4 \left(\frac{2-2d}{4}\right) + 4 \left(\frac{2-d}{2}\right) + 4(2) + 4(4+2d) + 4(8+8d) = 4 \left(\frac{49}{2}\right)$ $(2-2d) + 2(2-d) + 8 + (16+8d) + (32+32d) = 98$ $2-2d + 4-2d + 8 + 16+8d + 32+32d = 98$ Combine the constant terms: $2 + 4 + 8 + 16 + 32 = 62$. Combine the terms with $d$: $-2d - 2d + 8d + 32d = 36d$. The equation becomes: $62 + 36d = 98$ Subtract 62 from both sides: $36d = 98 - 62$ $36d = 36$ Divide by 36: $d = 1$

Step 5: Calculate the value of $a_4$. The question asks for $a_4$, which is the fifth term of the AGP ($T_5$). Using the formula for $T_5$ with $A=2$ and $d=1$: $T_5 = (A+2d) \cdot 4$ $T_5 = (2 + 2 \cdot 1) \cdot 4$ $T_5 = (2 + 2) \cdot 4$ $T_5 = 4 \cdot 4$ $T_5 = 16$ Therefore, $a_4 = 16$.

Common Mistakes & Tips

• Incorrect AGP Representation: Using a non-symmetric representation for the AP and GP terms can lead to more complex algebra. The symmetric approach with a known middle term is usually more efficient.
• Algebraic Errors: Be very careful with fractions and combining like terms. Multiplying by the least common denominator (LCD) at an early stage can simplify the process.
• Misinterpreting Notation: Ensure you understand which term the question is asking for. In this case, "$a_4$" refers to the fifth term of the sequence $a_1, a_2, 2, a_3, a_4$.

Summary The problem involves an arithmetico-geometric progression where a middle term is known and the sum of all terms is given. By representing the terms symmetrically and using the given information, we first determined the central term of the arithmetic progression ($A=2$). Then, by setting up and solving the equation for the sum of the AGP terms, we found the common difference of the arithmetic progression ($d=1$). Finally, we used these values to calculate the required fifth term of the AGP, $a_4$.

The final answer is $\boxed{16}$.