In arithmetic sequences, there is a number which is added or subtracted to obtain the next term - this is called the common difference and is shown by the coefficient of n when the nth term is found.
For example, the sequence 1,3,5,7… has a common difference of 2 and an nth term of 2n-1.
2(1)-1=1
2(2)-1=3
2(3)-1=5...
The ‘-1’ is the ‘0th term’, a constant to be added or subtracted and is found by subtracting the common difference from the first term (1-2).
When all the ‘light on’ numbers are odd, the nth term rule always has to have an odd ‘0th term’ and an even common difference.
The c.d has to be even because an odd common difference will generate a sequence with both even and odd terms (e.g. 3n = 3,6,9…).
The ‘0th term’ must be odd because if a sequence with an even c.d and 0th term will always have even terms (e.g. 2n+2 = 4,6,8…).
Some examples include:
2n-1 = 1,3,5,7…
-6n-9 = -15,-21,-27…
However, the common difference has to be an integer for this to hold true. For example, the sequence 0.6n+3 = 3.6,4.2,4.8…
Alternatively, the ‘0th term’ could be odd: 0.6n +3.5 = 4.1,4.7… as 0.6n will always generate an even number and 3.5 is odd. Even + odd = odd.
When all the ‘light on’ numbers are even, the nth term rule always has to have an even ‘0th term’ and an even common difference.
The c.d has to be even because an odd common difference will generate a sequence with both even and odd terms (e.g. 3n = 3,6,9…).
The ‘0th term’ must be even because if a sequence with an even c.d and odd 0th term will always have odd terms, as shown in the odd examples above.
Examples:
2n-2 =0,2,4,6…
-6n-8 = -14,-20,-26…
An even sequence can also be obtained if the common difference is a decimal and the 0th term is an integer, as displayed above (0.6n+3). In this case
However, if the ‘0th term’ is an odd decimal, then this will not hold true (as shown by 0.6n+3.5)
In sequences with a mixture of odd and even numbers, the nth term rule always has to have an odd common difference because an even common difference will result in a sequence with only odd or only even terms.
Some examples include:
3n-2 =1,4,7,10…
-5n-8 = -13,-18,-23….
When the terms are all multiples of a number, this must mean the common difference of the sequence and the coefficient of n in the nth term are the same. For example:
3,6,9,12… contains multiples of 3 and has an nth term of 3n.
4,8,12,16… contains multiples of 4 and has an nth term of 4n.
100,200,300…contains multiples of 100 and has an nth term of 100n.
Also, there must either be no 0th term, or a 0th which is also a multiple of the c.d of the sequence. This is proved by:
3n-1 (which does not work as -1 is not a multiple of 3, which is the c.d) = 2,5,8…
However, 3n-15 = -12,-9,-6… (as -15 is a multiple of 3, which is the c.d).
A sequence which has a last digit of 7 must have an nth term rule that has a common difference of a multiple of 10 and then a 0th term ending in either -3 or +7.
This is because a multiple of 10 +7 or -3 will always result in an integer ending in 7. Examples include:
10n-3 = 7,17,27…
100n+7= 107,207,307…
Alternatively, if the multiple of 10 as the c.d is negative, the 0th term must be +3 or -7. Examples:
-100n+3 = -97,-197,-297…
-10n-7= -17,-27,-37…