a geometric progression is one in which the terms are given by the formula:

a_n = ar^(n - 1) for n = 1,2,3,4,5...

where a_n is the nth term, a is the first term and r is the common ration

we are told that the first term of a particular geometric sequence is 5 and the second term is 4.8

now r is given by r = (a_{n+1})/(a_n)

in particular, r = (a_2)/(a_1) = 4.8/5 = 0.96

you are correct

The sum of the first n terms of a geometric sequence/progression is given by:ii) The sum of the firstnterms is greater than 124. Show that

0.96^n < 0.008,

and use logarithms to calculate the smallest possible value ofn.

S_n = a(1 - r^n)/(1 - r)

we are told that S_n > 124

=> a(1 - r^n)/(1 - r) > 124

=> 5(1 - (0.96)^n)/(1 - 0.96) > 124

=>5(1 - (0.96)^n)/(0.04) > 124

=> 5(1 - (0.96)^n) > 4.96 ..............i multiplied through by 0.04

=> 1 - (0.96)^n > 0.992 ..................i divided through by 5

=> -(0.96)^n > -0.008 ....................i subtracted 1 from both sides

=> (0.96)^n < 0.008 ...............................i multiplied through by -1, so i flipped the inequality sign.

Now we will use logs to find the smallest possible n.

0.96^n < 0.008

take log to the base 10 of both sides

=> log(0.96^n) < log(0.008)

=> nlog(0.96) < log(0.008)

=> n > log(0.008)/log(0.96) ...........note that log(0.96) is negative, so when i divided by it, i flipped the inequality sign

=> n > 118.27

so the smallest possible value for n is 119