If the 2nd and 5th term of a g.p are -6 and 48 respectively.Find the sum of the first four terms
Why did the geometric progression go to the doctor? Because it was feeling a bit negative and needed a check-up! Anyway, let's get down to business.
To find the common ratio (r) of the geometric progression, we can divide the 5th term by the 2nd term:
48 / -6 = -8
Now that we have the common ratio, we can find the first term (a) using the 2nd term and common ratio:
-6 = a * (-8)^1
-6 = -8a
Simplifying this equation gives us:
a = 6/8
a = 3/4
Now, to find the sum of the first four terms, we can use the formula for the sum of a geometric progression:
S = a * (1 - r^n) / (1 - r)
Plugging in the values, we get:
S = (3/4) * (1 - (-8)^4) / (1 - (-8))
Simplifying further:
S = (3/4) * (1 - 4096) / 9
S = (3/4) * (-4095) / 9
S = -12285 / 36
So, the sum of the first four terms is approximately -341.25.
To find the sum of the first four terms of a geometric progression (g.p.), we need to find the common ratio (r) first.
Given that the 2nd term of the g.p. is -6 and the 5th term is 48, we can use the formula for the n-th term of a g.p.:
nth term = a * r^(n-1)
We know that the 2nd term is -6, so we can substitute these values into the equation:
-6 = a * r^(2-1)
-6 = a * r^1
-6 = a * r
Similarly, we know that the 5th term is 48:
48 = a * r^(5-1)
48 = a * r^4
Now we have a system of equations with two unknowns (a and r):
-6 = a * r ...(1)
48 = a * r^4 ...(2)
To solve this system of equations, we can divide equation (2) by equation (1):
48 / -6 = (a * r^4) / (a * r)
-8 = r^3
Now we can substitute this value of r into equation (1) to find a:
-6 = a * r
-6 = a * -8
a = -6 / -8
a = 3/4
So, we have found the common ratio (r = -2) and the first term (a = 3/4).
To find the sum of the first four terms, we can use the formula for the sum of a g.p.:
Sum = a * (1 - r^n) / (1 - r)
Substituting the values, we get:
Sum = (3/4) * (1 - (-2)^4) / (1 - (-2))
= (3/4) * (1 - 16) / 3
= (3/4) * (-15) / 3
= -45/4
Therefore, the sum of the first four terms is -45/4 or -11.25.
Whoops sorry, thought ap but you said gp
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html
a , a r , a r^2 , a r^3 , a r^4
a r = -6
a r^4 = 48
so
-6 r^3 = 48
r^3 = -8
r = -2
then a = 3
continue as before
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
a+d = -6
a+ 4d = 48
---------------
-3 d = -54
d = 18
a + 18 = -6
a = -24
so
-24 , -6 , 12 , 30 , then 48 is next