The first term of GP exceed the second term by 2 and the sum to infinity is 15 find the series and the sum of first 5 term
first term --- a
second term ---- ar
a - ar = 2
a(1-r) = 2
sum to infinity = 15
a/(1-r) = 15
[a(1-r)][a/(1-r)] = 2(15)
a^2 = 30
a = ± √30
from a = ar+2
√30 = √30r + 2
r = (√30 - 2)/√30 = 1 - 2/√30
so we have √30, √30 + 2 , ....
sum(5) = a (r^5 - 1)/(r-1)
= ....
= .....
I will leave the cleanup up to you
To find the series and the sum of the first 5 terms of a geometric progression (GP), we need to use the given information.
Let's say the first term of the GP is 'a', and the common ratio is 'r'.
Given that the first term exceeds the second term by 2, we can write the second term as 'a/r' since the first term exceeds the second term by multiplying it by the common ratio (r). Therefore, we have the equation:
a - a/r = 2
Next, we know that the sum to infinity (S∞) of a GP is given by the formula:
S∞ = a / (1 - r)
Given that the sum to infinity is 15, we have the equation:
15 = a / (1 - r)
Now we have two equations with two unknowns (a and r). We can solve these equations simultaneously to find the values of 'a' and 'r'.
Solving the first equation for 'a', we get:
a - a/r = 2
Multiplying both sides by r:
ar - a = 2r
Adding 'a' to both sides:
ar = a + 2r
Factoring out 'a':
a(r - 1) = 2r
Dividing both sides by (r - 1):
a = 2r / (r - 1)
Now we substitute this value of 'a' in the second equation:
15 = (2r / (r - 1)) / (1 - r)
To simplify further, we multiply both sides by (r - 1)(1 - r) to get rid of the denominators:
15(r - 1)(1 - r) = 2r
Expanding and rearranging:
15(r^2 - 1) + 15r(1 - r) = 2r
15r^2 - 15 + 15r - 15r^2 = 2r
Combining like terms:
2r = 15
Dividing both sides by 2:
r = 15 / 2
r = 7.5
Now we can substitute the value of 'r' back into the equation for 'a':
a = 2r / (r - 1)
a = 2(7.5) / (7.5 - 1)
a = 15 / 6.5
a ≈ 2.3077
So, the common ratio (r) is 7.5, and the first term (a) is approximately 2.3077.
To find the series and the sum of the first 5 terms, we use the formula for a GP:
Series of the GP = a + ar + ar^2 + ar^3 + ...
Sum of the first n terms of the GP = a(r^n - 1) / (r - 1)
Substituting the values of 'a' and 'r', we have:
Series = 2.3077 + 2.3077 * 7.5 + 2.3077 * (7.5)^2 + 2.3077 * (7.5)^3 + 2.3077 * (7.5)^4
Sum of the first 5 terms = 2.3077(7.5^5 - 1) / (7.5 - 1)
Calculating these values, we get:
Series ≈ 2.3077 + 17.3077 + 128.8077 + 965.3077 + 7249.5577 ≈ 8363.29
Sum of the first 5 terms ≈ 2.3077(7.5^5 - 1) / (6.5)
≈ 2.3077(7890.62) / 6.5 ≈ 2778.17
Therefore, the series of the geometric progression is approximately 8363.29, and the sum of the first 5 terms is approximately 2778.17.