# Here is an arithmetic sequece find the 20 th term

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.

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

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 $32$.

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

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

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

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.

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.

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

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

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.

In Mathematica:

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


(* 32 *)

and the function derived is:

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