Question

The sum of 3 numbers in geometric progression is 38 and ther product is 1728. The middle number is

Answer 1

12

Answer 2

The problem asks us to find the middle number of three numbers in a geometric progression (G.P.) given their sum and product.

Let the three numbers in geometric progression be $\frac{a}{r}$, $a$, and $ar$, where $a$ is the middle term and $r$ is the common ratio.

We are given two pieces of information:

1. The sum of the numbers is 38. $\frac{a}{r} + a + ar = 38$

2. The product of the numbers is 1728. $(\frac{a}{r}) \times a \times (ar) = 1728$

Let's use the product information first, as it simplifies significantly: $a^3 = 1728$

To find the value of $a$, we need to calculate the cube root of 1728: $a = \sqrt[3]{1728}$

We can find the cube root by prime factorization or by recognizing common cubes: $10^3 = 1000$ $11^3 = 1331$ $12^3 = 1728$

So, $a = 12$.

Since we defined the three numbers as $\frac{a}{r}$, $a$, and $ar$, the middle number is $a$. Therefore, the middle number is 12.

We can also find the common ratio $r$ using the sum equation, though it's not required to answer the question about the middle number. Substitute $a=12$ into the sum equation: $\frac{12}{r} + 12 + 12r = 38$ Divide by 2: $\frac{6}{r} + 6 + 6r = 19$ Multiply by $r$: $6 + 6r + 6r^2 = 19r$ Rearrange into a quadratic equation: $6r^2 - 13r + 6 = 0$ Factoring the quadratic equation: $6r^2 - 9r - 4r + 6 = 0$ $3r(2r - 3) - 2(2r - 3) = 0$ $(3r - 2)(2r - 3) = 0$ This gives two possible values for $r$: $3r - 2 = 0 \implies r = \frac{2}{3}$ $2r - 3 = 0 \implies r = \frac{3}{2}$

If $a=12$ and $r=\frac{3}{2}$: The numbers are $\frac{12}{3/2} = 8$, $12$, $12 \times \frac{3}{2} = 18$. The sequence is 8, 12, 18. Sum = $8 + 12 + 18 = 38$. Product = $8 \times 12 \times 18 = 1728$.

If $a=12$ and $r=\frac{2}{3}$: The numbers are $\frac{12}{2/3} = 18$, $12$, $12 \times \frac{2}{3} = 8$. The sequence is 18, 12, 8. Sum = $18 + 12 + 8 = 38$. Product = $18 \times 12 \times 8 = 1728$.

In both valid cases, the set of numbers is $\{8, 12, 18\}$, and the middle number is 12.