The 2nd and 5th term of a geometric progression are - 6 & 48 find the sum of the first four term respectively
I think this method is for a. P, not gp
The correct one is
ar=-6
ar^4=48
r^3=-8
So r=-2
Sn=a(1-r^n)
- - - - - - - -
1-r
S4=3(1-(-2^4)
- - - - - - - -
1-(-2)
S4=3(1-16)
- - - - -
1+2
S4=3(-15)
- - - - -
3
S4=-15
That's the correct answer.
I think this method is for a. P, not gp
a+d = -6
a+4d = 48
so,
3d=54
d=18
so, a=-24
and the sequence is
-24 -6 12 30 48 ...
Well, it looks like this geometric progression decided to play a little hide-and-seek with us. Let's see, we know that the 2nd term is -6 and the 5th term is 48. Now, to find the common ratio, we can divide the 5th term by the 2nd term.
So, the common ratio is 48 / (-6) = -8.
Now, let's find the first term, shall we? We can do that by dividing the 2nd term by the common ratio. That gives us -6 / -8 = 3/4.
Now that we know the first term and the common ratio, we can find the sum of the first four terms of this sneaky geometric progression.
The formula for the sum of the first n terms of a geometric progression is given by:
S_n = a * (1 - r^n) / (1 - r),
Where S_n is the sum of the first n terms, a is the first term, and r is the common ratio.
So, plugging in the values, we get:
S_4 = (3/4) * (1 - (-8)^4) / (1 - (-8)).
Now, let's do some math magic and simplify this equation.
S_4 = (3/4) * (1 - 4096) / (1 + 8),
S_4 = (3/4) * (-4095) / 9,
S_4 = -1365 / 4.
And there you have it, the sum of the first four terms of this elusive geometric progression is -1365/4. So, even though this progression may be a bit tricky, we managed to catch it and find its sum!
To find the sum of the first four terms of a geometric progression, we need to determine the common ratio (r).
Given that the 2nd term (a2) is -6 and the 5th term (a5) is 48, we can use these values to find the common ratio.
The general formula for a geometric progression is:
an = a1 * r^(n-1)
Since a2 = -6, we can substitute these values into the formula:
-6 = a1 * r^(2-1)
Simplifying, we get:
-6 = a1 * r
Similarly, for the 5th term, we have:
48 = a1 * r^(5-1)
Simplifying, we get:
48 = a1 * r^4
Now, we can solve these two equations to find the values of a1 and r.
Dividing the second equation by the first, we get:
48 / -6 = (a1 * r^4) / (a1 * r)
Simplifying, we have:
-8 = r^3
Taking the cube root of both sides, we find:
r = -2
Now that we have the value of r, we can substitute it back into any of the two equations to find a1. Let's use the first equation:
-6 = a1 * (-2)
Simplifying, we get:
a1 = 3
Now we have determined that the first term (a1) is 3 and the common ratio (r) is -2.
The sum of the first four terms can be calculated using the formula for the sum of a geometric series:
S4 = a1 * (1 - r^4) / (1 - r)
Substituting the values of a1 and r, we have:
S4 = 3 * (1 - (-2)^4) / (1 - (-2))
Simplifying, we get:
S4 = 3 * (1 - 16) / (1 + 2)
S4 = 3 * (-15) / 3
S4 = -45
Therefore, the sum of the first four terms of the geometric progression is -45.