The second and fifth term of a G.P are 1 1/8 respectively.find the sum of infinity of the sequence
Find the values of x for which 1/3(2x+7)-1/5(1-4x)_<4+x
X-2y+1_>0
-8x-y-2>0
AAAaannndd the bot gets it wrong yet again!
ar=1
ar^4 = 1/8
so r^3 = 1/8, making r = 1/2
so a = 2 and
S = a/(1-r) = 2/(1 - 1/2) = 4
Depending on the value of y, we get -5/17 < x < 4
I suspect you have left something out.
To find the sum of an infinite geometric sequence, we need to determine the common ratio.
Let's start by finding the common ratio of the geometric progression (G.P) given that the second term is 1 and the fifth term is 1 1/8.
In a geometric progression, each term is obtained by multiplying the preceding term by a constant called the common ratio (r).
The second term, denoted as a2, is given as 1. Using the formula for the nth term of a geometric progression, we can express this as:
a2 = a1 * r^(2-1) => 1 = a1 * r
Similarly, the fifth term, denoted as a5, is given as 1 1/8 = 9/8. We can use the same formula:
a5 = a1 * r^(5-1) => 9/8 = a1 * r^4
Now, we can solve these two equations simultaneously to find the values of a1 and r.
From the first equation, we have a1 = 1/r. Substituting this into the second equation, we get:
9/8 = (1/r) * r^4
Simplifying the equation, we have:
9/8 = r^3
To get rid of the fraction, we can multiply both sides of the equation by 8:
9 = 8 * r^3
Now, let's find the cube root of both sides:
∛(9) = ∛(8 * r^3)
∛(9) = 2 * r
Dividing both sides by 2, we find:
r = ∛(9) / 2
Now that we have determined the common ratio (r), we can proceed to find the sum of the infinite geometric sequence.
The sum of an infinite geometric series can be calculated using the formula:
Sum = a1 / (1 - r)
Substituting the values we have:
Sum = 1 / (1 - ∛(9) / 2)
Hence, the sum of the infinite geometric sequence is 1 / (1 - ∛(9) / 2).
The sum of an infinite geometric progression is given by the formula:
S = a₁ / (1 - r)
where a₁ is the first term and r is the common ratio.
In this case, a₁ = 1 and r = 1/8.
Therefore, the sum of infinity of the sequence is:
S = 1 / (1 - 1/8) = 8/7