Let $$a_n be n^th the term of a decreasing infinite geometric progression. If $a_1+a_2+a_3=52$ and $a_1{a}_2+a_2{a}_3+a_3{a}_1=624$ , then the sum of this geometric progression is

The correct option is 54.

Step-by-Step Explanation:

Let the first term of the infinite geometric progression be $a_1=a$ and the common ratio be $r$.

The terms of the geometric progression can be expressed as:

$a_1=a$

$a_2=ar$

$a_3=a{r}^2$

Step 1: Use the first given condition.

We are given that the sum of the first three terms is 52:

$a_1+a_2+a_3=52$

Substitute the terms in terms of $a$ and $r$:

$a+ar+a{r}^2=52$

Factor out $a$:

$a(1+r+r^2)=52$     --- (Equation 1)

Step 2: Use the second given condition.

We are given that:

$a_1{a}_2+a_2{a}_3+a_3{a}_1=624$

Substitute $a_1=a$, $a_2=ar$, and $a_3=a{r}^2$:

$\left(a\right)\left(ar\right)+\left(ar\right)\left(a{r}^2\right)+\left(a{r}^2\right)\left(a\right)=624$

$a^{2}r+a^2{r}^3+a^2{r}^2=624$

Factor out $a^{2}r$:

$a^{2}r(1+r^2+r)=624$

$a^{2}r(1+r+r^2)=624$     --- (Equation 2)

Step 3: Divide Equation 2 by Equation 1.

$\frac{a^{2}r(1+r+r^2)}{a(1+r+r^2)}=\frac{624}{52}$

Simplifying both sides gives:

$ar=12$

$a=\frac{12}{r}$

Step 4: Find the value of $r$.

Substitute $a=\frac{12}{r}$ into Equation 1:

$\frac{12}{r}(1+r+r^2)=52$

Multiply both sides by $r$ and divide by 4:

$3(1+r+r^2)=13r$

$3+3r+3r^2=13r$

$3r^2-10r+3=0$

Factor the quadratic equation:

$(3r-1)(r-3)=0$

This gives $r=\frac{1}{3}$ or $r=3$.

Since the problem states that the geometric progression is decreasing and infinite, the common ratio must satisfy $0<r<1$ for the infinite sum to exist and converge. Therefore:

$r=\frac{1}{3}$

Step 5: Calculate the first term $a$ and the infinite sum $S_{\infty }$.

Now, calculate $a$:

$a=\frac{12}{1/3}=36$

The formula for the sum of an infinite geometric progression is:

$S_{\infty }=\frac{a}{1-r}$

Substitute the values of $a=36$ and $r=\frac{1}{3}$:

$S_{\infty }=\frac{36}{1-\frac{1}{3}}=\frac{36}{\frac{2}{3}}=36\times \frac{3}{2}=54$

Thus, the sum of this infinite geometric progression is 54.