what's the rule behind the sequence 1,25,100,225?
Answers
TRY THIS
find a missing number, first find a Rule behind the Sequence.
Sometimes we can just look at the numbers and see a pattern:
Example: 1, 4, 9, 16, ?
Answer: they are Squares (12=1, 22=4, 32=9, 42=16, ...)
Rule: xn = n2
Sequence: 1, 4, 9, 16, 25, 36, 49, ...
Did you see how we wrote that rule using "x" and "n" ?
xn means "term number n", so term 3 is written x3
And we can calculate term 3 using:
x3 = 32 = 9
We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" 25 wherever n is.
x25 = 252 = 625
How about another example:
Example: 3, 5, 8, 13, 21, ?
After 3 and 5 all the rest are the sum of the two numbers before,
That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:
3, 5, 8, 13, 21, 34, 55, 89, ...
Which has this Rule:
Rule: xn = xn-1 + xn-2
Now what does xn-1 mean? It means "the previous term" as term number n-1 is 1 less than term number n.
And xn-2 means the term before that one.
Let's try that Rule for the 6th term:
x6 = x6-1 + x6-2
x6 = x5 + x4
So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:
x6 = 21 + 13 = 34
Many Rules
One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works.
What is the next number in the sequence 1, 2, 4, 7, ?
Here are three solutions (there can be more!):
Solution 1: Add 1, then add 2, 3, 4, ...
So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc...
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, ...
(That rule looks a bit complicated, but it works)
Solution 2: After 1 and 2, add the two previous numbers, plus 1:
Rule: xn = xn-1 + xn-2 + 1
Sequence: 1, 2, 4, 7, 12, 20, 33, ...
Solution 3: After 1, 2 and 4, add the three previous numbers
Rule: xn = xn-1 + xn-2 + xn-3
Sequence: 1, 2, 4, 7, 13, 24, 44, ...