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Find the 20th term of this sequence. $1, 5, 4, 8, 7, 11, 10, 14.........$ it is like add 4 to the st term then subtract 1 i need to find a recursive formula that can help me get the 20th term. Help please.

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  • $\begingroup$ Do you need a recursive formula, or can you just write it out? $\endgroup$
    – Ian Coley
    Feb 10, 2015 at 20:44
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    $\begingroup$ oeis.org/A166517 $\endgroup$ Feb 10, 2015 at 20:45

8 Answers 8

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There are two arithmetic series in the given series. The numbers at odd position constitute the following series

$1,4,7,10,\cdots$

The numbers at even position constitute the following series:

$5,8,11,14,\cdots$

Since $20$ is an even number, the $20^{th}$ term will be the $10^{th}$ term of the second series. And it is $32$.

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Here's a single formula that works for all positive $n$:

$$a_n = \lfloor (3n/2) + 1 \rfloor + (-1)^n.$$

Then, $a_{20} = \lfloor 31 \rfloor + 1 = 32.$

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I know you didn't ask for an explicit rule but, you can also find that too...

$\\a_{2n}=5+3(n-1) \text{ or } a_{2n+1}=1+3n$

one of these gives you the even entries and the other gives you the odd entries.

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Every two steps, you add $3$. So after $20$ steps you'll have added $3$ ten times. This gets you the $21$st term. Now you can work out the $20$-th.

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Note that $$a_{2n}=a_{2n-1}+1\\ a_{2n-1}=a_{2n-2}+4,\quad n\ge 1$$

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The list:

0 1 2 3 4 5 6 7 8 9 10 ... 19 20

1 5 4 8 7 11 10 14 13 17 16 ... 32 31

The series: (4.5+3*n-2.5*(-1)^n)/2

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To write a recursive formula you need two parts: a starting point, and a rule for finding each term from the previous term. For this sequence, a recursive formula could look like:

$$\begin{array}{ll} a_1 = 1 \\ a_n = \left\{ \begin{array}{ll} a_{n-1} + 4 & \text{if } n \text{ is even} \\ a_{n-1} - 1 & \text{if } n \text{ is odd} \\ \end{array} \right. \end{array}$$

To use a recursive formula to find a specific term in a sequence you must find all of the terms before the specific term. I would just write out the sequence to the 20th term.

As other answers have noted, there are more efficient ways of finding the 20th term than using a recursive formula. For example using one of the explicit formulas given in other answers will give you the 20th term without needing to find any of the previous terms.

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In Mathematica:

FindSequenceFunction[{1, 5, 4, 8, 7, 11, 10, 14}, 20]

(* 32 *)

and the function derived is:

$\frac{1}{4} (-1)^n \left(3 (-1)^n (2 n+1)+5\right)$

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