Expert: Steve Holleran - 8/15/2008

Question
With a second number sequence we have a problem in that the records have been lost except for two pieces of information:-
The 12th number in the sequence is 15 and the sum of the 1st 5 numbers is 210.
Find the first 5 terms and give an expression for the n’th term.


Answer
Hi Nicky,

Okay, if the 12th term is 15, then using the formula for a specific term of an arithmetic progression, that tells us

              15 = a + 14d    where a = 1st term, d= common diff.

Also, using the formula for the sum of a specific number of terms, if the first five terms add to 210, you have

              210 = (5/2)(a + a5)

but a5 = a + 4d, so then  210 = (5/2)(a + (a+4d)

                         210 = (5/2)(2a + 4d)

then                      210 * (2/5) = 2a + 4d

                            84   = 2a + 4d

                            42  = a + 2d

Now take the two equations

                       a + 14 d = 15

                       a + 2d   = 42

and subtract.  You get    12d = - 27 do d = -2.25

Substitute this into a + 2d = 42:  a + (-4.5) = 42

              so                  a = 46.5

That makes the first five terms

    46.5,   44.25,  42,   39.75,  37.5

The nth term formula is

               a(n) = a + (n-1)d   so

               a(n) = 46.5 + (n-1)(-2.25)

which simplifies to a(n) = 48.75 - 2.25n


Hope this is what you wanted.
Steve