
If a, ${{a}^{2}}+2$ and ${{a}^{3}}+10$ are in GP, then find the value(s) of a.
Answer
618k+ views
Hint: Here, we may apply the concept that a GP is generated by multiplying each of its terms by a constant every time to get its successive terms and this constant is called the common ratio of the GP.
Complete step-by-step answer:
Complete step-by-step answer:
Since, the given terms of the GP are:
$a,\,{{a}^{2}}+2\,\,and\,{{a}^{3}}+10$.
We know that when the first term of the GP that is a is multiplied by the common ratio, we get the second term of the GP.
Let the common ratio of this GP be = r.
So, the common ratio will be:
$r=\dfrac{{{a}^{2}}+2}{a}...........(1)$
Similarly when we multiply the second term of the GP with the common ratio, then we will get the third term.
So, again for the common ratio we have:
$r=\dfrac{{{a}^{3}}+10}{{{a}^{2}}+2}...............(2)$
Since, both the equations (1) and (2) give a common ratio of the GP. So, we can equate both the equations and get:
$\dfrac{{{a}^{2}}+2}{a}=\dfrac{{{a}^{3}}+10}{{{a}^{2}}+2}$
On cross multiplication and further simplification we will get:
$ \left( {{a}^{2}}+2 \right)\times \left( {{a}^{2}}+2 \right)=\left( {{a}^{3}}+10 \right)\times a $
$ {{a}^{4}}+2{{a}^{2}}+2{{a}^{2}}+4={{a}^{4}}+10a $
$ 4{{a}^{2}}-10a+4=0 $
So, we have a quadratic equation in ‘a’. It means that by solving this equation we may have two values of a. So on dividing the equation so obtained by 2 we get:
$2{{a}^{2}}-5a+2=0...........(3)$
Now, we may solve this equation using factorization method:
$ 2{{a}^{2}}-4a-a+2=0 $
$ 2a\left( a-2 \right)-1\left( a-2 \right)=0 $
$ \left( 2a-1 \right)\left( a-2 \right)=0 $
So, we have:
$2a-1=0\,\,\,\,\,\,or\,\,\,a-2=0$
Hence, the two values of a are:
$a=\dfrac{1}{2}\,\,\,or\,\,\,a=2$
Recently Updated Pages
Master Class 11 Computer Science: Engaging Questions & Answers for Success

Master Class 11 Business Studies: Engaging Questions & Answers for Success

Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Maths: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Trending doubts
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

There are 720 permutations of the digits 1 2 3 4 5 class 11 maths CBSE

Discuss the various forms of bacteria class 11 biology CBSE

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE

State the laws of reflection of light

Explain zero factorial class 11 maths CBSE

