Animated Solution for Mathematics - Sequence and Series: If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2p)+\frac{1}{4^3}(3+3p)+\dots \infty$, then the value of $p$ is_____

The given infinite series is:

This expression represents an Arithmetico-Geometric Progression (AGP), where each term is the product of an Arithmetic Progression (AP) and a Geometric Progression (GP).

By observing the structure, we identify the components of the AP in the numerators: $3,3+p,3+2p,\dots$. Here, the first term is $a=3$ and the common difference is $d=p$.

The denominators form a GP: $1,\frac{1}{4},\frac{1}{4^2},\dots$. The common ratio for this sequence is $r=\frac{1}{4}$.

To find the sum of an infinite AGP where $|r|<1$, we utilize the standard formula:

Given that $r=\frac{1}{4}$, the condition $|r|<1$ is satisfied. Substituting our parameters $a=3$, $d=p$, and $r=\frac{1}{4}$ into the formula, we get:

First, we simplify the individual components of the equation. The first term evaluates as:

Next, we simplify the second term involving $p$:

Substituting these back into our master equation yields:

Subtracting $4$ from both sides results in $4=\frac{4p}{9}$. Solving for $p$, we find: