The second term of a GP is 81 and the fourth term is 9. Find,
(i) the common ratio,
(ii) the sum of the first terms,
(iii) the sum to infinity.
2nd term is 81 ---- ar = 81
4th term is 9 ----- ar^3 = 9
divide second equation by the first
r^2 = 9/81 = 1/9
r = ± 1/3
so ar = 81
a(1/3) = 81
a = 243
ii) you have a type, but now that you have a and r, whatever the question should be simple
iii) sum of infinite
=a/(1-r) = 243/(1-1/3)
= 243/(2/3)
= 243(3/2) = 364.5
i want to answer this question
good answer
To find the common ratio of a geometric progression (GP), you can divide any term by the previous term. Let's use the given information to find the common ratio:
(i) The second term of the GP is 81, and the fourth term is 9. We can set up the following equation to find the common ratio (r):
81 / 9 = r
Simplifying the equation, we get:
r^2 = 9
Taking the square root of both sides, we get:
r = 3 or r = -3
So, the possible values for the common ratio are 3 or -3.
(ii) To find the sum of the first "n" terms of a GP, you can use the formula:
Sn = a * (r^n - 1) / (r - 1)
Where Sn is the sum of the first "n" terms, a is the first term, and r is the common ratio.
We know the second term is 81, and we need to find the sum of the first n terms, so we substitute the values into the formula:
81 = a * (r^2 - 1) / (r - 1)
We can substitute the possible values of r (3 and -3) to check if the equation holds for both:
For r = 3:
81 = a * (3^2 - 1) / (3 - 1)
81 = a * (9 - 1) / 2
81 = a * 8 / 2
81 = 4a
a = 81/4
For r = -3:
81 = a * ((-3)^2 - 1) / ((-3) - 1)
81 = a * (9 - 1) / (-4)
81 = a * 8 / (-4)
81 = -2a
a = -81/2
So, the first term (a) can be either 81/4 or -81/2, depending on the value of the common ratio (r).
(iii) To find the sum to infinity (S∞) of a converging geometric progression (|r| < 1), you can use the formula:
S∞ = a / (1 - r)
Based on the possible values of the common ratio, let's calculate the sum to infinity for each case:
For r = 3:
S∞ = (81/4) / (1 - 3) = (81/4) / (-2) = -81/8
For r = -3:
S∞ = (-81/2) / (1 - (-3)) = (-81/2) / 4 = -81/8
So, in both cases, the sum to infinity (S∞) is -81/8.