Basicaly, I need help on this sequence, as my homework is due in tomorrow. One question on sequences is:
1
3
11
67
I'm supposed to find out what comes next. Can anyone help? Even if you tell me how you got the answer, that would be a great help
Basicaly, I need help on this sequence, as my homework is due in tomorrow. One question on sequences is:
1
3
11
67
I'm supposed to find out what comes next. Can anyone help? Even if you tell me how you got the answer, that would be a great help
Hello, snakeyster!
I see a pattern . . .I need help on this sequence: .$\displaystyle 1,\,3,\,11,\,67,\,\cdots$
. . $\displaystyle \begin{array}{ccccc} a_1 & = & 1 & & \\
a_2 & = & {\color{red}2}(a_1)\,{\color{red}+\,1} & = & 3 \\
a_3 & = & {\color{red}4}(a_2)\,{\color{red}-\,1} & = & 11 \\
a_4 & = & {\color{red}6}(a_3)\,{\color{red}+\,1} & = & 67\end{array}$
I bet the next one is:
. . $\displaystyle \begin{array}{ccccc} a_5 & = & {\color{red}8}(a_4)\,{\color{red}-\,1} & = & 535\end{array}$
The general term seems to be: .$\displaystyle a_n\;=\;2(n-1)(a_{n\text{-}1}) + (\text{-}1)^n$ . for $\displaystyle n \geq 2$
You made no mistake, one can chose the next term to have an arbitary
value and find a polynomial that will take all the given values and our arbitary
next value.
Your cubic does give the first four terms, and so is as valid a rule for the
next term as any other, and I suspect it involves no more arbitary constants
than does Soroban's
RonL
HellO!
RonL and Dan are, of course, absolutely correct.
A sequence of numbers is meaningless unless there is a stipulation,
. . a promise that that the sequence "continues in a similar manner",
. . that there is indeed a "reasonable" pattern to the terms.
I've posted the following before, but I enjoy showing it off.
Find the next term of the sequence: .$\displaystyle \bf{1,\,3,\,5,\,7,\,\cdots}$
Answer: 8
Justification
I was using this function: .$\displaystyle f(n) \;=\;-\frac{1}{24}\left(n^4 - 10n^3 + 35n^2 - 98x + 48\right) $
"That's unfair!" you say?
Okay, here's a better explanation . . .
The sequence: .$\displaystyle 1,\,3,\,5,\,7,\,8,\,\cdots$ is the sequence of natural numbers
. . whose English names contain the letter "e".