The first term of GP is 4if the common ratio is 2 find the greatest number of terms that will give a sum less than 40
The sum of the first n terms of a geometric progression can be given by the formula:
S = a(1 - r^n) / (1 - r)
where S represents the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 4 and r = 2. Substituting these values into the formula, we get:
S = 4(1 - 2^n) / (1 - 2)
Simplifying further, we have:
S = 4(1 - 2^n) / (-1)
S = -4(2^n - 1)
We want the sum to be less than 40, so we can set up the inequality:
-4(2^n - 1) < 40
Dividing both sides by -4, we have:
2^n - 1 > -10
Adding 1 to both sides, we get:
2^n > -9
Since the number of terms (n) should be a positive integer, we can conclude that the greatest number of terms that will give a sum less than 40 is when n = 4.
So, the greatest number of terms is 4.
In a GP the sum of the 2 and 3 is 4and the sum of 3 and 4 terms is 2 find the term and the common ratio
Let's assume that the first term of the GP is 'a' and the common ratio is 'r'.
The sum of the 2nd and 3rd terms is 4, so we have:
a * r + a * r^2 = 4
The sum of the 3rd and 4th terms is 2, so we have:
a * r^2 + a * r^3 = 2
We can rewrite these equations as follows:
a * r * (1 + r) = 4 [1]
a * r^2 * (1 + r) = 2 [2]
Dividing equation [1] by equation [2], we get:
(a * r * (1 + r)) / (a * r^2 * (1 + r)) = 4 / 2
Simplifying, we have:
r / r^2 = 2
r^(-1) = 2
1/r = 2
r = 1/2
Substituting the value of r = 1/2 into equation [1], we have:
a * (1/2) * (1 + (1/2)) = 4
a * (1/2) * (3/2) = 4
3a/4 = 4
Multiplying both sides by 4/3, we get:
a = (4/3) * (3/4)
a = 1
Therefore, the first term (a) is 1 and the common ratio (r) is 1/2.
Mutiso's salary is 12000 pounds p.a his salary increases by 10 percent annually find the total amount he will have earned in six years
To find the total amount Mutiso will have earned in six years, we can use the formula for the future value of an annuity with annual compounding:
Future Value = Present Value * (1 + interest rate)^number of periods
In this case, Mutiso's salary is the present value, which is £12000, and his salary increases by 10% annually, so the interest rate is 10% or 0.1.
Plugging in the values, we can calculate the future value:
Future Value = £12000 * (1 + 0.1)^6
Future Value = £12000 * (1.1)^6
Future Value = £12000 * 1.771561
Future Value = £21259.73
Therefore, Mutiso will have earned a total amount of £21259.73 in six years.
The sum of the first three terms of geometric sequence is 14 if the common ratio is -3 find the sum of the first five terms
To find the sum of the first five terms of a geometric sequence, we can use the formula:
S = a * (1 - r^n) / (1 - r)
where S represents the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, we know that the sum of the first three terms is 14, so we can set up the equation:
14 = a * (1 - r^3) / (1 - r)
We are given that the common ratio is -3, so we substitute -3 for r:
14 = a * (1 - (-3)^3) / (1 - (-3))
Simplifying the expression within the parentheses, we get:
14 = a * (1 - 27) / (1 + 3)
14 = a * (1 - 27) / 4
14 = a * (-26) / 4
Multiplying both sides by 4, we have:
56 = -26a
Dividing both sides by -26, we get:
a = -56 / -26
a = 28/13
So, the first term (a) is 28/13 and the common ratio (r) is -3.
To find the sum of the first five terms, we plug these values into the sum formula:
S = (28/13) * (1 - (-3)^5) / (1 - (-3))
Simplifying further, we have:
S = (28/13) * (1 - 243) / (1 + 3)
S = (28/13) * (-242) / 4
S = (28/13) * (-121)
S = -616/13
Therefore, the sum of the first five terms is -616/13 or approximately -47.3846.