800 Followers Problem #2

Algebra Level 4

The sequence $\{a_n\}$ is defined by $a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \dfrac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.$ The sequence $\{b_n\}$ is defined by $b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \dfrac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.$

Find $\dfrac {b_{32}}{a_{32}}$ .

The answer is 561.

2 solutions

Gautam Sharma
Apr 17, 2015

Sequence a is very easy easy to figure out and we obtain the general term for sequence a: $a_n =n!$

For series b after calculating $b_3 ,b_4$ the pattern can be observed and it is:

$b_3=3\times 4 \times 5$

$b_4=3 \times 4 \times 5 \times6$

Hence

$b_n=3 \times 4 \times 5 .............. (n+2)$

Hence $\frac{b_{32}}{a_{32}}$ :

$\frac{b_{32}}{a_{32}}=\frac {3 \times 4 \times 5 .............. (34)}{32!}$

$\Rightarrow \boxed {33\times 17 =561}$

Rohit Sachdeva
Apr 16, 2015

For a-series We can observe the pattern & see that $a_{n}=n!$

For b-series we have:

$b_{0}=0!*1$

$b_{1}=1!*3$

$b_{2}=2!*6$

$b_{3}=3!*10$

So we have n! x (a level-2 A.P with common difference 2,3,4,.....33)

$b_{32}=32!*(1+2+3+.....33)=32!*561$

$b_{32}/a_{32}=561$

800 Followers Problem #2

The answer is 561.

2 solutions

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