If $\alpha ,\beta $ are roots of the equation $375{x^2} - 25x - 2 = 0$ and ${S_n} = {a^n} + {\beta ^n},$ then $\mathop {\lim }\limits_{n \to \infty } \mathop \sum \limits_{r = 1}^n {S_r}$ is equal to

JEE MATHEMATICS MCQ: QUADRATIC EQUATIONS

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\alpha ,\beta

are roots of the equation

375{x^2} - 25x - 2 = 0

and

{S_n} = {a^n} + {\beta ^n},

then

\lim\limits_{n \to \infty } \sum \limits_{r = 1}^n {S_r}

is equal to

View answer and explanation

Correct option: C

\frac{{29}}{{348}}

Option C (

\frac{{29}}{{348}}

) is correct because by finding the sum of the infinite geometric series formed by the roots and using the relationships between the roots and coefficients, the limit simplifies to the given fraction.

More Questions from QUADRATIC EQUATIONS

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The solution of the equation

\log _7 \log _5(\sqrt{x+5}+\sqrt{x})=0

is If

\alpha ,\beta

are the roots of

{x^2} - 3x + a = 0,a \in R

and

\alpha < 1 < \beta ,

then If

\alpha ,\;\beta

be the roots of the equation

{x^2} - px + q = 0

and

{\alpha _1},{\beta _1}

be the roots of the equation

{x^2} - qx + p = 0,

then the equation whose roots are

\frac{1}{{{\alpha _1}\beta }} + \frac{1}{{\alpha {\beta _1}}}\mathrm{and}\;\frac{1}{{\alpha {\alpha _1}}} + \frac{1}{{\beta {\beta _1}}},

is If

{x^2} + 4ax + 2 > 0

for all values of

x

, then

a

lies in the interval If

f\left( x \right) = a{x^2} + bx + c\;

and

\;f\left( { - 1} \right) < 1,f\left( 1 \right) > - 1,f\left( 3 \right) < - 4\;

and

\;a \ne 0,

then Root(s) of the equation

9{x^2} - 18\left| x \right| + 5 = 0

belonging to the domain of definition of the function

f\left( x \right) = \log \left( {{x^2} - x - 2} \right)

is (are)