arithmetic and geometric sequences

Question

Given the terms T10 = 3/512  and T15 = 3/16384  of a geometric sequences, find the exact value of the term T30 of the sequences

 

 

Answer 1

In geometric sequence a n = ar^n-1

Here r is common ratio.

Given T10 is ar^9 = 3/512

T15 is ar^14 = 3/16384

Now divideT15 by T10.

ar^14/ar^9 = (3/16384)/(3/512)

r^5 = 1/32

r^5 = 1/2^5

r^5 =(1/2)^5

r = 1/2

Substitute the r value in T10.

a(1/2)^9 = 3/512

a(1/2^9) = 3/512

a(1/512) = 3/512

a = (3/512)512

a = 3

Now T30 = ar^29 = 3*(1/2)^29

= 3*1/2^29

= 3/2^29

= 3/536870912