Question

Find the four numbers in G.P, whose sum is $85$ and the product is $4096$.
A.$64,16,4,1$
B.$64,16,40,1$
C.$64,19,4,1$
D.$67,16,4,1$

Answer 1

Using the concept of G.P as let the first term be a and general ratio be r, then nth term can be given as ${{\text{T}}_{\text{n}}}{\text{ = a}}{{\text{r}}^{{\text{n - 1}}}}$. So from the given concept form all the terms and then solve it using the condition given in the question.

Complete step by step answer:

Let the four terms of a G.P be ,
${\text{a,ar,a}}{{\text{r}}^{\text{2}}}{\text{,a}}{{\text{r}}^{\text{3}}}$
Now, as per the given that sum of all the terms is $85$

$$\Rightarrow {\text{a + ar + a}}{{\text{r}}^{\text{2}}}{\text{ + a}}{{\text{r}}^{\text{3}}} = 85 \\\ \Rightarrow {\text{a(1 + r + }}{{\text{r}}^{\text{2}}}{\text{ + }}{{\text{r}}^{\text{3}}}{\text{) = 85}} \\\$$

And the product of the four terms is $4096$

$$\Rightarrow {\text{a(ar)(a}}{{\text{r}}^{\text{2}}}{\text{)(a}}{{\text{r}}^{\text{3}}}) = 4096 \\\ \Rightarrow {\text{(}}{{\text{a}}^4}{{\text{r}}^6}{\text{) = 4096}} \\\$$

Calculating the factors of

$$\Rightarrow {\text{4096 = }}{{\text{a}}^{\text{4}}}{{\text{r}}^{\text{6}}}{\text{ = }}{{\text{2}}^{{\text{12}}}} \\\ \Rightarrow {{\text{a}}^{\text{4}}}{{\text{r}}^{\text{6}}}{\text{ = 1}}{\text{.}}{{\text{4}}^{\text{6}}} \\\ \Rightarrow {\text{a = 1,r = 4}} \\\$$

And so from this we can conclude that the terms are,

$${\text{a = 1,r = 4}} \\\ {\text{1,4,16,64}} \\\$$

Hence, option (a) is the correct answer.

Note: In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Properties :
If all the terms of G.P are multiplied or divided by the same non-zero constant then the sequence remains in G.P with the same common ratio.
The reciprocals of the terms of a given G.P. form a G.P