Geometric Sequence help please?

Question

A geometric series is such that the sum of its first two terms is 28 and the sum of its fourth and fifth terms is 756.

a) Write down two equations in involving a and r

b) factorise both equations

Answer 1

We know that geometric series is a,ar,ar^2,ar^3,ar^4....ar^n

Here a is first term of series and r iscommon ratio.

Formula for n th term in gemetric series is a(r^n-1)

In given problem sum of first two terms is 28.

a+ar = 28

Take common out a from above expression.

a(1+r) = 28  -----> (1)

Sum of fourth and fifth terms is 756.

ar^3+ar^4 = 756

ar^3(1+r) = 756 -----> (2)

From (1) and (2) equations

1+r = 28/a and 1+r = 756/ar^3

28/a = 756/ar^3

cross multiplication.

28ar^3 = 756a

Cancell the a.

28r^3 = 756

r^3 = 756/28

r^3 = 27

Aplly cube root on each side.

r = 3

Common ratio is 3.

a(1+3) = 28

4a = 28

a = 7

Then the given series wouldbe 7,21,63,189,567,...