First slide

Theory of expressions

Question

If $α$ and $β$ are the roots of the equation ${ax}^{2} + bx + c = 0 and S_{n} = α^{n} + β^{n}, then$ ${aS}_{n + 1} + {bS}_{n} + {cS}_{n - 1}$ is equal to

Moderate

A

0
B

abc
C

a+b+c
D

None of these

Solution

Given, $α$ and $β$ are the roots of equation ${ax}^{2} + bx + c = 0$ and $S_{n} = α^{n} + β^{n}$
$∴ α + β = - \frac{b}{a} and αβ = \frac{c}{a}$
Now, $\begin{aligned} S_{n + 1} = α^{n + 1} + β^{n + 1} \\ = α^{n + 1} + β^{n + 1} + α^{n} β + β^{n} α - α^{n} β - β^{n} α \\ = α^{n} (α + β) + β^{n} (α + β) - αβ (α^{n - 1} + β^{n - 1}) \\ = (α + β) (α^{n} + β^{n}) - αβ (α^{n - 1} + β^{n - 1}) \\ = - \frac{b}{a} S_{n} - \frac{c}{a} S_{n - 1} \end{aligned}$
$\begin{aligned} \Rightarrow S_{n + 1} = \frac{- {bS}_{n} - {cS}_{n - 1}}{a} \\ ∴ {aS}_{n + 1} + {bS}_{n} + {cS}_{n - 1} = 0 \end{aligned}$ .

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App