Question

Find five numbers in GP such that the product is 243 and sum of second and fourth number is 10

Answer 1

Answer:

The first five numbers of GP are: 1/3, 1, 3, 9, 27 or 27, 9, 3, 1, 1/3.

Step-by-step explanation:

Consider the provided information.

Let the five numbers in GP are:

$\frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2$

The product is 243.

$\frac{a}{r^2} \times \frac{a}{r} \times a \times ar \times ar^2=243$

$a^5=243$

$a=3$

Thus the value of a is 3. Now, calculate the value of r.

The sum of second and fourth number is 10.

The second number is $\frac{a}{r}$ and fourth number is ar.

$\frac{a}{r}+ar=10$

$a+ar^2=10r$

$ar^2-10r+a=0$

Substitute the value of a = 3 in above equation.

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

$3r^2-9r-r+3=0$

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

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

$r-3=0$ or $3r-1=0$

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

Thus, the value of r is 3 or 1/3. There are two different values of r, that means there is two possible GP.

Hence, the first five numbers of GP are: 1/3, 1, 3, 9, 27 or 27, 9, 3, 1, 1/3.

Answer 2

Step-by-step explanation:

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