Maths questions - series and sequences.

[acroviak]

I am stuck on a few maths questions on series / sequences

q1. The third term of a geometric sequence is 81 and the 6th is 24.
(a) Show that the common ratio is 2/3

q2.The first three terms of a geometric sequence are a,b,c. Each term represents an increase of p percent on the preceeding term.

(a) i) Show that the common ratio is (1+(p/100))

ii) It is given that a = 2000. Express b and c in terms of p.

thanks for any help

[Buenchico]

Typing mathematical notation is always a pain in the @rse, but I'll have a go.

A1: Let the common ratio be K.

From the information given, and the definition of a common ratio, 81 x K x K x K = 24

<=> K cubed = 24/81 = 8/27

<=> K = cube root of 8/27 = (cube root of 8)/(cube root of 27) = 2/3


A2(i): From the information given b = a + (ap/100).

From the definition of a common ration, K = b/a = (a + (ap/100))/a = 1 + p/100


A2(ii): From the definition of a common ratio, b =aK (where C is the common ratio).

Given that a = 2000, and substituting for K from the previous answer:

b = 2000 (1 + p/100) = 2000 + 2000p/100 = 2000 + 20p

Further (from the definition of a common ratio), c = bK.

Thus, c = 20(100 + p) (1 + p/100) = 20 (100 + p + p + p~squared/100) = 20 (100 +2p + p~squared/100) = p~squared/5 + 40p + 2000


Chris

[Buenchico]

Oops!

I've just spotted a minor typo.

I originally used 'C' to stand for 'common ratio' but then I saw that 'c' appeared in your question, so I changed 'C' to 'K'. Unfortunately, I omitted to change one occurrence of 'C'.

So A2(ii) should start thus:

A2(ii): From the definition of a common ratio, b =aK (where K is the common ratio)

Sorry about that.

Chris