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How to find the nth terms?

Example: Find the ratio and the seventh terms of the sequences
2/9, 2/3, 2, 6, 18

First step:
Divide the successive pairs of the terms as long as they are next to each other

(2/9) / (2/3) = 2/9 / 3/2 = 3
2 / (2/3) = 2/1 / 3/2 = 3
6/2 = 3
18/6 = 3

Second step:
Find the ratio

The ratio is always 3, so r = 3

Third step:
Find the sixth terms and seventh terms

a6 = 18 x 3 = 54
a7 = 54 x 3 = 162

Final step:
Rewrite your answer

Ratio = 3
Seventh terms = 162

Now done with geometric sequences. Thank You :)

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How to find the nth terms?

Example: Find the nth terms of the sequence
3,5,7,9

First step:
Find the difference of the sequences
The differences are always the number behind the n
So the nth term begins with 2n

Second step:
Worked out what the previous number would be or you can say find the number before 3.
= 1,3,5,7,9

Third step:
Add number 1 into the nth term formula
= 2n + 1

Final step:
Now check your answer if it correct. Check it with n = 1,2,3,4 and + 1

n = 1          2 x 1 + 1 = 3
n = 2          2 x 2 + 1 = 5
n = 3          2 x 3 + 1 = 7
n = 4          2 x 4 + 1 = 9

The answer is the same with the sequences given which are 3,5,7,9. So we're done with linear sequences

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1. The following arrangement of numbers is known as the Pascal's Triangle
2. Each row begins and ends with the number 1 and it has one number more than the previous row.
3. To obtain the numbers in the next row, add the two numbers that are immediately above it in the previous row.

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0,1,1,2,3,5,8,13,21,34,55,89,......
1. In the above sequence of numbers, it begins with 0,1,1 and the numbers that follow are obtained by adding two numbers before it
2. This sequence of numbers is called Fibonacci Numbers

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How to find the nth term?

Note: The answer will always be in this form  = an³ + bn² + cn+ d

0, 12, 10, 0, -12, -20

First step:
Find the difference from the 1st term until u get a constant difference.
1st difference = 12, -2, -10, -12
2nd difference =-14, -8, -2
3rd difference =-6, 6

Second step:
Divide the third difference by 6 and you will get the value of (a)
6÷6=1                 So the first term of the nth term is n3

Third step:
Substitute the number 1 to 5 into n3 and find the difference between the original sequence and n³
n= 1, 2, 3, 4, 5
n³= 1, 8, 27, 64, 125, 216
s-n³=-1, 4, -17, -64, -137, -236

Fourth step:
Find the difference in s-n3
1st difference = 5, -21, -47, -73
2nd difference =--26, -26, -26
-26÷2= -13²

Fifth step:
Work out s-(n³-13² )

(n³-13² )= -12, -44, -90, -144, 200
s-(n³-13² )= 12, 56, 100, 144, 188
Common difference = 44

Final step:
12+ (n-1) 44 = 44n-44 = 44n + 32
Final answer = n³-13²+44n + 32

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How to find the nth term?

Note: The answer will always be in this form  = an² + bn + c

4, 15, 32, 55, 84

First step:
Find the difference from the 1st term until u get a constant difference.
1st difference = 11, 17, 23, 29
2nd difference = 6, 6, 6

Second step:
Divide the second difference by 2 and you will get the value of (a)
6÷2=3                   So the first term of the nth term is 3n²

Third step:
Substutute the number 1 to 5 into 3n²
n= 1, 2, 3, 4, 5
3n²=3, 12, 27, 48, 75

Fourth step:
Take the 3n² and the original number sequence (4, 15, 32, 55, 84) and work out the nth term of these numbers
3n²=3, 12, 27, 48, 75
Original sequence:4, 15, 32, 55, 84
Differences= 1, 3, 5, 7, 9
Now the nth term of these differences (2, 4, 6, 8, 10) is 2n -1.

So b = 2 and c = -1

Final step:
Write down your final answer in the form of an² + bn + c.
=3n² + 2n -1

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• All even numbers end with the digits 0,2,4,6 or 8.
• All odd numbers end with the digits 1,3,5,7 or 9.

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1. The list of even  numbers:0,2,4,6,8,10,12,14,16,18,20
2. Every even number is divisible by 2
3. The list of odd number :1,3,6,7,9,11,13,15,17,19,
4. Every odd number is not divisible by 2