Question

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression equals:

Answer 1

Here in this question we have been asked to find the common ratio of the given geometric progression. To answer this question we will assume that the first term is $a$ and the common ratio is $r$ and use the given statement “each term is equal to the sum of the next two terms”.

Complete step by step answer:
Now considering from the question we have been asked to find the common ratio of the given geometric progression.
From the basic concepts of progressions, we know that the ${{n}^{th}}$ term of the geometric progression is given as ${{a}_{n}}=a{{r}^{n-1}}$ .
Let us suppose that the first term is $a$ and the common ratio is $r$ then the first three terms will be given as $a,ar,a{{r}^{2}}$ .
Now by considering the given statement “each term is equal to the sum of the next two terms” we will have $a=ar+a{{r}^{2}}$ .
Now by simplifying the equation we will have $1=r+{{r}^{2}}$ .
Now we will solve this quadratic equation using the formula for finding the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ will be given as $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
By using the formula we will have the solutions for the equation ${{r}^{2}}+r-1=0$ are $\dfrac{-1\pm \sqrt{5}}{2}$.
Now we have been given that the terms are positive therefore the ratio should be positive. Because the first term is positive as it is one of the terms. Any term in the progression can be expressed as a multiple of first term and common ratio.
Hence we can conclude that the common ratio of the given progression is $\dfrac{-1+\sqrt{5}}{2}$ .

Note: While answering questions of this type we should verify every condition given in the question. If someone had ignored the condition that all terms are positive then they will end up having two different possible answers which is a wrong conclusion.