in a GP,the product of the 2nd and 4th terms doubled the 5th term and sum of the first four term is 80.Find the GP.
1A,1B,
To find the geometric progression (GP), we'll assume the common ratio as 'r' and solve step by step.
Let's assume the first term of the GP as 'a'.
The second term would be 'ar' (since it's a GP),
The third term would be 'ar^2',
The fourth term would be 'ar^3',
And the fifth term would be 'ar^4'.
According to the given information:
1) The product of the 2nd and 4th terms is twice the 5th term:
(ar)(ar^3) = 2(ar^4)
2) The sum of the first four terms is 80:
a + ar + ar^2 + ar^3 = 80
Now, let's solve these equations step by step:
1) (ar)(ar^3) = 2(ar^4)
ar^4 * ar^3 = 2(ar^4)
ar^7 = 2ar^4
Dividing by 'ar^4' on both sides:
r^3 = 2
Taking the cube root of both sides:
r = ∛2 = 1.26 (approximately)
2) a + ar + ar^2 + ar^3 = 80
a(1 + r + r^2 + r^3) = 80
a(1 + 1.26 + 1.26^2 + 1.26^3) = 80
Calculating the expression inside the brackets:
(1 + 1.26 + 1.26^2 + 1.26^3) ≈ 4.015
Dividing both sides by '4.015':
a ≈ 80 / 4.015
a ≈ 19.95 (approximately)
Therefore, the geometric progression (GP) is approximately:
a, ar, ar^2, ar^3, ar^4
19.95, 25.14, 31.73, 40.03, 50.42