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.
Thefactor
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,
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
Example:
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 ofsuch
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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