suppose-that-E-9-4-k-2-n-a-k-2-is-given-let-a-k-a-k-1-a-k-1-1-and-a-1-1-a-2-2-a-3-6-

Question Number 198813 by mnjuly1970 last updated on 24/Oct/23

s u p p o s e t h a t :

E = \sqrt{9 + 4 (\underset{k = 2}{\sum^{n}} a_{k}^{2})}

i s g i v e n

l e t a_{k} = a_{k - 1} . (a_{k - 1} + 1)

a n d a_{1} = 1, a_{2} = 2, a_{3} = 6,

a_{4} = 42, a_{5} = 1806 a n d e t c

f i n d t h e v a l u e o f E = ?

Answered by witcher3 last updated on 24/Oct/23

a_{k} - a_{k - 1} = a_{k - 1}^{2}

\Leftrightarrow \underset{k = 3}{\sum^{n + 1}} (a_{k} - a_{k - 1}) = \underset{k = 2}{\sum^{n}} a_{k}^{2}

\Leftrightarrow a_{n + 1} - a_{2} = \underset{k = 2}{\sum^{n}} {(a_{k})}^{2}

E = \sqrt{9 + 4 (a_{n + 1} - a_{2})} = \sqrt{9 + 4 (a_{n + 1} - 2)}

= \sqrt{1 + {4 a}_{n + 1}} = \sqrt{1 + {4 a}_{n} (a_{n} + 1)} = \sqrt{{({2 a}_{n} + 1)}^{2}}

=∣ {2 a}_{n} + 1 ∣= {2 a}_{n} + 1, a_{n} > 0

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