Maths Qs (Sequences & Series)

sleepyescapade

Hope I can post this here! Just wondering is there a particular formula I could use to work out the following questions:

(sample exam questions)

An arithmetic sequence has first term −2 and sixth term 13 . The common difference is
(a) 11/5 , (b) 13/6 , (c) 3 .

An arithmetic sequence has first term 6 and seventh term −3. The common difference is
(a) 3/2 , (b) –3/2 , (c) –6/7


A geometric sequence has first term −48 and fourth term 6. The common ratio is (a) 16 , (b) −1/2 , (c) –1/3 .


A geometric sequence has first term 72 and fourth term –8/3. The common ratio is
(a) –1/3 , (b) 1/3 , (c) –1/6 .


The formulae in my notes are

for arithmetic : ai = a1 + ( i - 1 ) d
geometric: ai = a1 + i-1

however i dont really understand how they work

Michael Collins

Yep there is a formula for working it out. For an arithmetic series you are just adding some number, called the "common difference", onto the previous term.

For example in the first question you are given the first term, -2, and told that the 6th is 13. So if we say d is the common difference we have:

1st term = -2
2nd term = -2 + d
3rd term = -2 + d + d = -2 + 2d

and so on until we get to the 6th term:

6th term = -2 + 5d

which we are told equals 13 so:

-2 + 5d = 13
5d = 15
d = 3.

So your first formula, ai = a1 + (i-1)d, was the one to use because a1 is the first term (-2 in this case) and i is the term you want, (6 in this case), and d again is the common difference (that you want to find out) so putting this into the formula you get:

ai = a1 + (i-1)d
a6 = -2 + (6-1)d = 13
-2 + 5d = 13
5d = 15
d = 3

(same as above).

Does this help for the first part anyway?

Fremen

Your geometric sequence formula's borked

To get the next term of a geometric sequence, you multiply the previous term by the common ratio.
Like michael collins was doing above,
a1 = a1

a2 = a1*r

a3 = a2*r = a1*r*r

a4 = a3*r = a2*r*r = a1*r*r*r

There's a pattern here: the Nth term is a1 multiplied by r, (N - 1) times.
This can be expressed as
ai = a1*(r^(i-1))

sleepyescapade

Thank you both very much