IB Maths Notes: Arithmetic Sequences
An arithmetic sequence is a series of numbers such that to get the next number in the sequence we add a number to the last term. We add the SAME number each time. For example
4, 9, 14, 19, 24 is an arithmetic sequence because we add 5 to each term to get the next term. The general form for the nth term in a geometric sequence is:
an=a+(n-1)d, where a is the first term and d is the difference between any two successive terms.
The (n-1) reflects the fact that to get the 1st term we don't have to add anything: only from the 1st term do we start adding things.
When we add up n terms, we write down an expression like,
Sn=a+(a+d)+(a+2d)+(a+3d)+(a+4d)+................+(a+(n-2)d)+(a+(n-1)d)
By writing this backwards we obtain,
Sn=(a+(n-1)d)+(a+(n-2)d)+.................................+(a+2d)+(a+d)+a
We can now add the two sequences, getting 2Sn on the left hand side and altogether n terms all the same, 2a+(n-1)d on the right hand side, so
We may be asked: The 3rd
term of an arithmetic sequence is 9 and the 5th term is
17. Find the first term, the common difference and the smallest value
of n such that
and
Now solve the simultaneous equations
(1)
(2)
Sub
into
(1)
Solve
Non integer or negative values of n are not allowed
here, because we are considering only the natural numbers, so
