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:
whereis
the first term andis
the difference between any two successive terms.
Thereflects
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,
By writing this backwards we
obtain
We can now add the two
sequences, gettingon
the left hand side and altogether n terms all the same,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
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