# Finding the nth term of decreasing linear sequences (math help)

Updated on February 8, 2018 ## Nth Term Of A Decreasing Sequence Video

Finding the nth term of a decreasing linear sequence can by 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              (1st row)

(2nd row)

5              3              1              -1            -3            (3rd 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              (1st row)

-2            -4            -6            -8            -10          (2nd row)

5              3              1              -1            -3            (3rd 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.

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              (1st row)

(2nd row)

-9            -13          -17          -21          -25          (3rd row)

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

1              2              3              4              5              (1st row)

-4            -8            -12          -16          -20          (2nd row)

-9            -13          -17          -21          -25          (3rd 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.

• Find the ninth term of the sequence. 3​, 1​, -3​, -9​, -17​?

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.

• Find the ninth term of the sequence. 27, 25, 23, 21, 19?

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.

• 156 , 148 , 140 , 132 which term will be the first to be negative ?

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.

• How do you calculate the second difference of a quadratic sequence without the first term?

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.

## Related

Submit a Comment
• Maxwell

7 years ago

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

• 8 years ago

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

• elfin

8 years ago

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

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

• Park Min Shin

8 years ago

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

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

• For cody111

8 years ago

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

8 years ago

lol.

• cody111

8 years ago

hello,

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

2 5 10 17 16

thanks lots:)

• Zachman

8 years ago

• AUTHOR

Mark

8 years ago from England, UK

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

8 years ago

I ready knew this i needed help

With creating a sequence

With the nth term as:

12-2n

• AUTHOR

Mark

9 years ago from England, UK

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

9 years ago

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 ;)

9 years ago

Somebody Helpp me pleaseeeee :) xxxxxxxxxxxxxxxxxxxx

• nokaa :) xx

9 years ago

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

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

working