the-sequence-a-1-a-2-a-3-satisfies-the-relation-a-n-1-a-n-a-n-1-for-n-gt-1-given-that-a-20-6765-and-a-18-2584-what-is-a-16-

Question Number 87648 by john santu last updated on 05/Apr/20

the sequence a_{1}, a_{2}, a_{3}, \dots satisfies

the relation a_{n + 1} = a_{n} + a_{n - 1}, for

n > 1 . given that a_{20} = 6765 and

a_{18} = 2584 what is a_{16}

Commented by john santu last updated on 05/Apr/20

a_{20} = a_{19} + a_{18}

= a_{18} + a_{17} + a_{18}

= 3 \times a_{18} - a_{16}

\Rightarrow a_{16} = 3 \times a_{18} - a_{20}

\Rightarrow a_{16} = 3 \times 2584 - 6765 = 987

Commented by john santu last updated on 05/Apr/20

so for this question

x^{2} - x - 1 = 0

x = \frac{1 \pm \sqrt{5}}{2}

a_{n} = A {(\frac{1 + \sqrt{5}}{2})}^{n} + B {(\frac{1 - \sqrt{5}}{2})}^{n}

Commented by mr W last updated on 05/Apr/20

y e s

Commented by john santu last updated on 05/Apr/20

thank you sir