Answer
Verified
421.8k+ 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
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE