How do find the first four terms of an arithmetic sequence when given the first term (A1), An, and Sn?example: Find the first four terms. A1=10...

The two formulas you need to
know:

Find any term in an arithmetic sequence: An = A1 +
(n-1)d

where A1 = is the first term, n is the number in the
sequence, and d is the common difference.

Find the sum of
an arithmetic sequence: Sn = 0.5n(A1+An).

To solve d, we
need to know n for the An=67.  To do this we can use the sum
equation:

Sn = 0.5n(A1+An)
680 =
0.5n(10+67)
n = (2*680)/(10+67) = 17.66

This
equation is not contingent on n being a whole number, and so it is irrelevant that it is
not a whole number in this case.  It is customary for it to be a whole number, but the
equation does not fail if is not.

It is my opinion that
rounding it makes your final answer less accruate.  It is possible to have a value in
between two values in an arthimetic series, and as such, insisting that the 3.2nd term
in a sequence cannot exist is just bad math; values exist between numbers.  If they did
not, interpolating, and extrapolating would be moot
concepts.

You can still use the data given to find the
terms in the sequence, regardless of whether the value 67 is an nth term or a between
nth term.

Now, to determine d we can use the first
equation:

An = A1 + (n-1)d
67 = 10 +
(17.66-1)d
d = (67-10)/(16.66) =
3.42

Now:

A1 = 10
A2
= 10 + (2-1)(3.42) = 13.42
A3 = 10 + (3-1)(3.42) = 16.84
A4 = 10 +
(4-1)(3.42) = 20.26