the first term of a gp is twice its common ratio. find the sum of the first two terms of the progression if its sum to infinity is 8
We are told that
a = 2r
sum of all terms = a/(1-r)
2r/(1-r) = 8
2r = 8 - 8r
10r = 8
r = 4/5
then a = 8/5
sum of first two terms = a + ar
= 8/5 + (8/5)(4/5) = 50/5 = 10
check for sum of all terms to be 8
sum(all terms) = (8/5) / (1 - 4/5)
= (8/5) / 1/5)
= 8
Is it correct
We are told that
a = 2r
sum of all terms = a/(1-r)
2r/(1-r) = 8
2r = 8 - 8r
10r = 8
r = 4/5
then a = 8/5
sum of first two terms = a + ar
= 8/5 + (8/5)(4/5)
=8/5 + 32/25
Final answer =72/25
The answer is not correct the final answer is 72/25
The answer is very wrong the correct answer is either 72/25 or 2 22/25
To solve this problem, we need to first establish a relationship between the terms of the geometric progression (GP). Let's assume that the first term of the GP is "a", and the common ratio is "r".
According to the given information, the first term is twice the common ratio. So we can write the equation:
a = 2r ... (Equation 1)
The sum of an infinite geometric progression is given by the formula:
Sum = a / (1 - r)
Substituting the given value of the sum to infinity and using Equation 1, we have:
8 = a / (1 - r) ... (Equation 2)
Now, we need to find the sum of the first two terms of the progression. This sum can be calculated by summing the first term "a" and the second term "ar":
Sum of first two terms = a + ar
To find this sum, we can substitute Equation 1 into the above equation:
Sum of first two terms = 2r + 2r^2
Now, let's solve for "r" using Equation 2:
8 = (2r) / (1 - r)
To simplify the equation, we can multiply both sides by (1 - r):
8(1 - r) = 2r
Expanding the left side:
8 - 8r = 2r
Adding 8r to both sides:
8 = 10r
Dividing both sides by 10:
r = 8/10
Simplifying:
r = 0.8
Now, substitute this value of "r" into the equation for the sum of the first two terms:
Sum of first two terms = 2r + 2r^2
= 2(0.8) + 2(0.8)^2
= 1.6 + 2(0.64)
= 1.6 + 1.28
= 2.88
Therefore, the sum of the first two terms of the progression is 2.88.