A g.p has 8 terms.If the sum of first 4 terms is 45 and the sum of the last 4 term is 720, find the common ratio and the first term.
S4 = a (r^4-1)/(r-1) = 45
S8 = a(r^8-1)/(r-1) = 45+720 = 765
divide and you have (r^8-1)/(r^4-1) = 765/45 = 17
r^8-1 = (r^4-1)(r^4+1)
so, r^4+1 = 17
r^4 = 16
r = ±2
Now just find a
How did you get -+2
I didn't understand the 2nd line
Op
Well, let's see if I can calculate this one without pulling a prank.
To find the common ratio (r) and the first term (a) for a geometric progression (g.p.), we need to use the sum formula for a g.p.:
Sn = a * (1 - r^n) / (1 - r),
where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.
Given that the sum of the first 4 terms (S4) is 45, we have:
S4 = a * (1 - r^4) / (1 - r) = 45.
Also, the sum of the last 4 terms (S8) is 720:
S8 = a * (1 - r^8) / (1 - r) = 720.
Now that we have two equations, we can solve for a and r. Let's do some calculations!
S8 = 720 ⇒ a * (1 - r^8) / (1 - r) = 720.
S4 = 45 ⇒ a * (1 - r^4) / (1 - r) = 45.
Oh wait, I just realized I can't solve this with humor alone. Let me help you find a solution in a more serious way. Give me a moment, please.
To find the common ratio (r) and the first term (a) in a geometric progression (g.p), we can use the formulas for the sum of the first n terms of a g.p:
S(n) = a(1 - r^n) / (1 - r)
Where:
S(n) represents the sum of the first n terms,
a is the first term,
r is the common ratio,
and n is the number of terms.
We are given that the g.p has 8 terms, so n = 8.
We can set up two equations using the given information:
Equation 1: S(4) = 45
This means the sum of the first 4 terms is 45.
Equation 2: S(8) = 720
This means the sum of all 8 terms is 720.
Let's substitute the values into the equations and solve them simultaneously.
Equation 1:
S(4) = a(1 - r^4) / (1 - r) = 45
Equation 2:
S(8) = a(1 - r^8) / (1 - r) = 720
Now, let's solve these equations.
Using Equation 1, we have:
45 = a(1 - r^4) / (1 - r)
Rearranging the equation, we get:
45(1 - r) = a(1 - r^4)
Expanding the equation, we get:
45 - 45r = a - ar^4
And further simplifying, we have:
45r^4 - 45r + a - 45 = 0
Similarly, using Equation 2, we have:
720 = a(1 - r^8) / (1 - r)
Rearranging the equation, we get:
720(1 - r) = a(1 - r^8)
Expanding the equation, we get:
720 - 720r = a - ar^8
And further simplifying, we have:
720r^8 - 720r + a - 720 = 0
So, we have two equations:
45r^4 - 45r + a - 45 = 0
720r^8 - 720r + a - 720 = 0
We can solve these equations simultaneously to find the values of r and a.