First slide

Theory of expressions

Question

If $α, β$ are the roots of $x^{2} - ax + b = 0$ and if $α^{n} + β^{n} = V_{n},$ then

Moderate

A

$V_{n + 1} = {aV}_{n} + {bV}_{n - 1}$
B

$V_{n + 1} = {aV}_{n} + {aV}_{n - 1}$
C

$V_{n + 1} = {aV}_{n} - {bV}_{n - 1}$
D

$V_{n + 1} = {aV}_{n - 1} - {bV}_{n}$

Solution

Multiplying $x^{2} - ax + b = 0 by x^{n - 1}$
$x^{n + 1} - {ax}^{n} + {bx}^{n - 1} = 0 . . . (i)$
$α, β are roots of x^{2} - ax + b = 0,$ therefore they will satisfy (i) also $α^{n + 1} - {aα}^{n} + {bα}^{n - 1} = 0 . . . (ii)$
and $β^{n + 1} - {αβ}^{n} + {bβ}^{n - 1} = 0 . . . (iii)$
adding (ii) and (iii)
$(α^{n + 1} + β^{n + 1}) - a (α^{n} + β^{n}) + b (α^{n - 1} + β^{n - 1})$ =0
or $V_{n + 1} = {aV}_{n} - {bV}_{n - 1} (Given α^{n} + β^{n} = V_{n})$

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