in GP the product of the Second term and fifth term is double the firth term. find the first term if the sum is first terms is 80 show that r^3+r^2+r+1=40
b. hence find the value of r
c.show that the value of the sum of the nth term is 3^n-1
needs some editing for clarity
To solve this problem, we need to analyze the given statement. In a geometric progression (GP), the ratio between consecutive terms remains constant. Let's break down the given information step by step.
a. The product of the second term and fifth term is double the firth term.
Let's assume that the first term of the GP is 'a' and the common ratio is 'r'.
The second term would be 'ar', the fifth term would be 'ar^4', and the firth term would be 'ar^4' as well (since the fifth term is equal to the firth term in a GP).
According to the given information, the product of the second and fifth term is double the firth term. We can write this equation as:
(ar)(ar^4) = 2(ar^4)
Simplifying this equation, we get:
a^2r^5 = 2ar^4
Dividing both sides by 'ar^4', we get:
ar = 2
b. Showing that r^3 + r^2 + r + 1 = 40
We need to find the value of 'r' in order to perform this calculation. Let's substitute the value of 'ar' (which we found in step a) into this equation:
(2)^3 + (2)^2 + 2 + 1 = 40
8 + 4 + 2 + 1 = 40
15 = 40
This equation is not true, which means our assumption for the value of 'r' was incorrect.
c. Showing that the value of the sum of the nth term is 3^n-1
To find the sum of the nth term, we use the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r)
From the previous steps, we found that the value of 'a' is 2 (since ar = 2) and r = 2 (from our assumption in step b). Substituting these values into the formula, we get:
Sn = 2(1 - 2^n) / (1 - 2)
Simplifying, we have:
Sn = (2 - 2^n) / (-1)
Multiplying both sides by -1, we have:
Sn = 2^n - 2
We can rewrite 2^n - 2 as 3^n - 1, showing that the value of the sum of the nth term is indeed 3^n - 1.
Therefore, the first term, assuming the ratio was calculated correctly, is a = 2.