# Finding the nth Term of Decreasing Linear Sequences

*Mark, a math enthusiast, loves writing tutorials for stumped students and those who need to brush up on their math skills.*

## How to Find the nth Term of a Decreasing Linear Sequence

Finding the nth term of a decreasing linear sequence can be harder to do than increasing sequences, as you have to be confident with your negative numbers. A decreasing linear sequence is a sequence that goes down by the same amount each time.

Make sure that you can find the nth term of an increasing linear sequence before you try decreasing linear sequences. Remember, you are looking for a rule that takes you from the position numbers to the numbers in the sequence!

**Example 1**

Find the nth term of this decreasing linear sequence.

5 3 1 -1 -3

First of all, write your position numbers (1 to 5) above the sequence (leave a gap between the two rows)

1 2 3 4 5 (1^{st} row)

(2^{nd} row)

5 3 1 -1 -3 (3^{rd} row)

Notice that the sequence is going down by 2 each time, so times your position numbers by -2. Put these into the 2nd row.

1 2 3 4 5 (1^{st} row)

-2 -4 -6 -8 -10 (2^{nd} row)

5 3 1 -1 -3 (3^{rd} row)

Now try to work out how you get from the numbers on the 2nd row to the numbers on the 3rd row. Do this by adding on 7.

So to get from the position numbers to the term in the sequence, you have to times the position numbers by -2 and then add on 7.

## Read More From Owlcation

Hence the nth term = -2n + 7.

**Example 2**

Find the nth term of this decreasing linear sequence

-9 -13 -17 -21 -25

Again, write your position numbers above the sequence (remember to leave a gap)

1 2 3 4 5 (1^{st} row)

(2^{nd} row)

-9 -13 -17 -21 -25 (3^{rd} row)

Notice that the sequence is going down by 4 each time, so times your position numbers by -4. Put these into the 2^{nd} row.

1 2 3 4 5 (1^{st} row)

-4 -8 -12 -16 -20 (2^{nd} row)

-9 -13 -17 -21 -25 (3^{rd} row)

Now try to work out how you get from the numbers on the 2nd row to the numbers on the 3rd row. Do this by taking away 5.

So to get from the position numbers to the term in the sequence, you have to times the position numbers by -4 and then take away 5.

Hence the nth term = -4n - 5.

## Questions & Answers

**Question:** 15,12, 9, 6 what is nth term?

**Answer:** This sequence is going down in 3's so compare is to the negative multiplies of 3 (-3,-6,-9,-12).

You will need to add 18 to each of these numbers to give the numbers in the sequence.

So the nth term of this sequence is -3n + 18.

**Question:** What is the nth term for 20,17,14,11?

**Answer:** -3n + 23 is the answer.

**Question:** Find the ninth term of the sequence. 3, 1, -3, -9, -17?

**Answer:** The first differences are -2, -4, -6, -8, and the second difference are -2.

Therefore since half of -2 is -1 the first term will be -n^2.

Subtracting -n^2 from the sequence gives 4,5,6,7,8 which has nth term n + 3.

So the final answer is -n^2 + n + 3.

**Question:** How do you calculate the second difference of a quadratic sequence without the first term?

**Answer:** The first term doesn't have to be given, all that is required to calculate the second difference is that there are three consecutive terms.

**Question:** 156 , 148 , 140 , 132 which term will be the first to be negative ?

**Answer:** Its probably easier just to continue the sequence until you reach the negative numbers.

The sequence is decreasing by 8 each time.

156, 148, 140, 132, 124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4, -4 ...

So this will be the 21st term in the sequence.

**Question:** Find the ninth term of the sequence. 27, 25, 23, 21, 19?

**Answer:** The first differences are -2, so compare the sequence with the multiples of -2 (-2,-4,-6,-8,-10)

You will have to add 29 to these multiples to give the numbers in the sequence.

So the nth term is -2n + 29.

**Question:** What is the nth term of the sequence { -1, 1, -1, 1, -1 }?

**Answer:** (-1)^n.

**Question:** If the nth term of a sequence is 45 - 9n what is the 8th term?

**Answer:** First multiply 9 by 8 to give 72.

Next work out 45 - 72 to give -27.

**Question:** -1,1,-1,1,-1 nth term. How do I solve this?

**Answer:** (-1)^n.

**Question:** 3/8 of number is 12, what is the number?

**Answer:** 12 divided by 3 is 4, and 4 times 8 is 32.

## Comments

**Aarya Ray Chaudhuri** on March 14, 2020:

"little" late to the party, but through "independent" research I have discovered that you can also use the equation:

-Dn+ (a+D)

This will only work for sequences that decrease as the nth term increases.

Normally for sequences that increase, like 1,2,3,4....to aleph null, I use the equation :

Dn+(a-D)

If I were to take the sequence 1,3,5... I would observe that "a" is equal to 1 i.e. the first term, and "D" is 2, which I got from subtracting the second term with the first, i.e. 3-1.

I tried using the same equation for a decreasing linear sequence, and observed that the first term works, but after the first the numbers increase, which went against my decreasing sequene. Par exemple:

Lets take the term 32,25,18,11,4...

I see that the "a" variable is equal to 32, and "D" is equal to "32-25", which is 7. If I were to put this into the first sequence, i.e. "Dn+(a-D)", I would get:

nth=7n+25.

For the first term, I would observe

nth= (7*1)+(25), which is equal to 32, meaning the first term is correct. BUT, if I did for the second term(do the maths yourself im sorry), I would get 39, which is WRONG. After 3 minutes of giving it some thought, i devised the equation "-Dn+(a+D)". Eager to try it out, I went in.

D, as we saw before, is equal to 7, and "a" is equal to 32. Puttng this into the equation, I saw:

-7n+(32+7)

-7+39

For the first term, we see that:

(-7*1)+39= -7+39, which is equal to 32. If I do this for the second term, I observe:

(-7*2)+39= -14+39, which is 25.

Therefore, ergo, hence, in conclusion, I think that "-Dn+(a-D)" is a viable equation for a decreasing linear term.

GOODBYEEEEEEEEEEEEE MOOOONMEEEENNNNNNNN

**Maxwell** on February 27, 2012:

Try to put chartlfow and graphical representations.That will help understand linear sequences.hanks.

**nada {eygption}** on December 13, 2011:

i will tell u how to find the 100th term you take your number and times by 99

**elfin** on October 27, 2011:

yuzz...i really need help for a head aching problem of sequence:

wots the nth term of 54,53,50,43....???????

**Park Min Shin** on October 19, 2011:

Anneyeonghaseyo! I have a test tomorrow wish me luck! :)

btw the first word was saying "Hello".. Annyeong

**For cody111** on October 19, 2011:

The answer for that sequence in N squared + 1... because 1x1 (position 1) 1 + 1 = 2... 2(position)x2=4 +1=5.. hope u understand.. luv u! :)

**amy** on September 14, 2011:

lol.

**cody111** on September 08, 2011:

hello,

i just needed to know about non-linear equations it,s really confusing.

please help me on this sequence's nth term:

2 5 10 17 16

thanks lots:)

**Zachman** on August 23, 2011:

Thanks this was helpful!

**Mark (author)** from England, UK on January 08, 2011:

1s term is 12 - 2 x 1 = 12 - 2 = 10

2nd term is 12 - 2 x 2 = 8

3rd term is 12 - 2 x 3 = 6

4th term is 12 - 2 x 4 = 4

5th term is 12 - 2 x 5 = 2

So the sequence is 10,8,6,4,2 etc...

**No-One** on January 08, 2011:

I ready knew this i needed help

With creating a sequence

With the nth term as:

12-2n

**Mark (author)** from England, UK on September 02, 2010:

Hi jay,

You can use the formula:

nth term = a + (n-1)d.

a is the first number in the sequence and d is the common difference of the sequence.

**jay** on September 02, 2010:

ok but what if u get a queston which says find the formula for a sequence and the numbers are in aposition such as 100th 101th 103rd 104th 105th and so on.

eg:

find the formula of this sequence when the 100th is 258 101th is 260 102nd is 262 103rd is 264 104th 266 and 105th is 268th

good luck

**ex oh ex oh ;) ** on March 27, 2010:

Somebody Helpp me pleaseeeee :) xxxxxxxxxxxxxxxxxxxx

**nokaa :) xx** on March 27, 2010:

Thaanxxxxxxx a lot for the information, i didn't get it !!!

oh lala, hola, ge no sE* jo.!!