Animated Solution for Mathematics - Sequence and Series: Let $a_1,a_2,a_3,\dots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1{a}_9+a_2{a}_4{a}_9+a_5+a_7$ is equal to

Imagine you are standing on the edge of a vast, unfolding mathematical landscape. You see a sequence of numbers, each one born from the last by a constant, rhythmic multiplication. This is the Geometric Progression (GP), a structure of pure, exponential beauty.

We define our sequence as $a_1,a_2,a_3,\dots$ where the general term is $a_n=a{r}^{n-1}$. We are given that the sequence is increasing and all terms are positive, which implies that our common ratio $r$ must satisfy $r>1$.

The problem provides the condition that the product of the fourth and sixth terms is nine:

Substituting the general form, we get $(a{r}^3)(a{r}^5)=9$, which simplifies to $a^2{r}^8=9$. Notice that $a^2{r}^8$ is equivalent to $(a{r}^4{)}^2$.

Since $a{r}^4$ is the fifth term $a_5$, we have $a_5^2=9$. Because all terms are positive, we conclude that $a_5=3$.

Next, we are given that the sum of the fifth and seventh terms is twenty-four:

Substituting $a_5=3$, we find $3+a_7=24$, which implies $a_7=21$. We can express $a_7$ in terms of $a_5$ as follows:

Substituting our known values, we get $3r^2=21$. Dividing both sides by three, we find that $r^2=7$.

We now evaluate the target expression: $a_1{a}_9+a_2{a}_4{a}_9+a_5+a_7$. Let us break this down into its constituent parts:

First, consider the product $a_1{a}_9$:

Second, consider the product $a_2{a}_4{a}_9$:

Finally, we recall that $a_5+a_7=24$. Summing these components together, we obtain:

The final result of the expression is $60$.