Find-the-nth-term-of-the-sequence-0-3-8-15-24-

Question Number 206532 by necx122 last updated on 17/Apr/24

F i n d t h e n t h t e r m o f t h e s e q u e n c e

0, 3, 8, 15, 24, \dots \dots

Answered by Berbere last updated on 17/Apr/24

I m p o s s i b l

t h e r e a r e [i n f i n i t y w a y

n^{2} - 1; n \in Z_{+} o n e o f t h e m

Answered by BaliramKumar last updated on 18/Apr/24

0 \underset{+ 3}{\underset{⏟}{}} 3 \underset{+ 5}{\underset{⏟}{}} 8 \underset{+ 7}{\underset{⏟}{}} 15 \underset{+ 9}{\underset{⏟}{}} 24 \dots \dots \dots

a = 3 d = 5 - 3 = 2

S_{n - 1} = \frac{n - 1}{2} [2 \times 3 + (n - 1 - 1) \times 2]

S_{n - 1} = (n - 1) [3 + (n - 1 - 1)]

S_{n - 1} = (n - 1) (n + 1) = n^{2} - 1

U_{1} + S_{n - 1} = U_{n} = 0 + n^{2} - 1

U_{n} = n^{2} - 1

Commented by necx122 last updated on 18/Apr/24

w o w! T h i d i s s o c l e a r . T h a n k y o u .

Answered by Frix last updated on 17/Apr/24

I get the very nice

u_{n} = (n - 1)! - \frac{(n - 5)}{24} (9 n^{3} - 37 n^{2} + 70 n - 48)

or if you {don}^{'} t like n!

u_{n} = \frac{1}{2} (2^{n} - \frac{n^{4}}{12} + \frac{n^{3}}{2} + \frac{n^{2}}{12} + \frac{3 n}{2} - 4)

Commented by BaliramKumar last updated on 18/Apr/24

how to find ?

Commented by Frix last updated on 18/Apr/24

Just be creative .

Commented by BaliramKumar last updated on 19/Apr/24

not true for n \geq 6

Commented by mr W last updated on 19/Apr/24

o n l y t h e f i r s t 5 n u m b e r s a r e g i v e n .

t h e n u m b e r s u p o n t h e 6^{t h} c a n b e

a n y n u m b e r s . t h a t m e a n s s u c h a

q u e s t i o n m a y h a v e i n f i n i t e l y m a n y

s o l u t i o n s . t h e r e i s n o r e a s o n f o r

y o u t o s a y t h a t t h e 6^{t h} m u s t b e 35 .

Commented by Frix last updated on 19/Apr/24

Usually we search for a polynome :

n numbers given \Rightarrow polynome of degree n - 1

In the following cases this gives the red

next numbers :

0, 1, 2

0, 1, 1, 0

0, 1, 1, 2, 6

0, 1, 1, 2, 3, 0

0, 1, 1, 2, 3, 5, 16

0, 1, 1, 2, 3, 5, 8, 0

\dots

Think about it!

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