Question 1541
CBSE Class 11 Mathematics · Sequences and Series · Special series and applications
The roots a,b of x²−3x+p=0 and the roots c,d of x²−12x+q=0 occur consecutively in a geometric progression. Prove (q+p):(q−p)=17:15.
Solution to Question 1541
Complete workingTranslate the conditions into a sequence equation, then retain every admissible real solution.
\[\begin{aligned}\text{Write }a &= A,\ b\\&= Ar,\ c\\&= Ar^2,\ d\\&= Ar^3\end{aligned}\]
\[\begin{aligned}a+b &= A(1+r)\\&= 3\end{aligned}\]
\[\begin{aligned}c+d &= Ar^2(1+r)\\&= 12\Rightarrow r^2\\&= 4\end{aligned}\]
\[\begin{aligned}p &= ab\\&= A^2r,\qquad q\\&= cd\\&= A^2r^5\\&= p r^4\\&= 16p\end{aligned}\]
\[\begin{aligned}\frac{q+p}{q-p} &= \frac{16p+p}{16p-p}\\&= \frac{17}{15}\end{aligned}\]
The final value satisfies the original recurrence, sum, product or financial condition.
Final answer
\[(q+p):(q-p)=17:15\]