Question

Let α and β be the roots of the equation 5x² + 6x – 2 = 0. If S_n = αⁿ + βⁿ, n = 1,2,3...., then :

(1) 5S₆ + 6S₅ = 2S₄ 

(2) 5S₆ + 6S₅ + 2S₄ = 0 

(3) 6S₆ + 5S₅ + 2S₄ = 0 

(4)  6S₆ + 5S₅ = 2S₄

Answer 1

Answer is (1) 5S₆ + 6S₅ = 2S₄ 

α and β are roots of 5x² + 6x – 2 = 0

5α² + 6α – 2 = 0

5αⁿ⁺² + 6αⁿ⁺¹ – 2αⁿ = 0 …(1)

(By multiplying αⁿ)

Similarly 5βⁿ⁺² + 6βⁿ⁺¹ – 2βⁿ = 0 …(2)

By adding (1) & (2)

5S_n+2 + 6S_n+1 – 2S_n = 0

For n = 4

Answer 2

\(\alpha \:and\:beta\:are \:roots\:of\:\)\(\mathsf{5x^2+6x-2}\)

\(\mathsf{5\alpha^2+6\alpha-2=0}\)

→\(\mathsf{multiply \:a^n}\)

\(\mathsf{5\alpha^{n+2}+6\alpha^{n+1}-2=0}\)

\(\mathsf{similarly\: 5\beta^{n+2}+6\beta^{n+1}-2=0}\)

→\(\mathsf{adding\:two\:equations}\)

\(\mathsf{5S_{n+2}+6S_{n+1}-2S_n=0}\)

→\(\mathsf{here\: S\:is\:\alpha+\beta} \)

\(\mathsf{by\:taking\:n=4\:we\:get}\\ \mathsf{5S_6+6S_5=2S_4}\)