A GP is such that the 3rd term is nine times the 1st term while the 2nd term is one - twenty fourth of the 5th term .Find it's fourth term.
Well, it seems like this geometric progression is playing a bit of a hide-and-seek game with us. Let's try to uncover its fourth term with some clownish calculations!
Since the third term is nine times the first term, we can write:
a₃ = 9a₁
Now, the second term is one-twenty fourth of the 5th term:
a₂ = (1/24)a₅
Okay, now let's use the formula for a geometric progression to find the ratio (r):
r = a₂ / a₁
And since a₃ = 9a₁, we know the ratio squared should be equal to 9:
r² = 9
Solving for r, we find:
r = ±3
Since we're all about positivity, let's choose r = 3. Now we can find the fourth term:
a₄ = a₃ * r
Substituting the known values:
a₄ = 9a₁ * 3
a₄ = 27a₁
So, the clownishly calculated fourth term of this geometric progression is 27 times the first term. Now it's up to you to reveal the exact value of the first term, and then we can determine the fourth term together!
To find the fourth term of the geometric progression (GP), we first need to determine the common ratio (r) of the GP.
Let's assume that the first term of the GP is 'a' and the common ratio is 'r'.
Given that the 3rd term is nine times the 1st term, we can write the equation:
a * r^2 = 9a
This equation implies that r^2 = 9, which means r = 3 or r = -3.
As for the 2nd term being one-twenty fourth of the 5th term, we can write the equation:
a * r = (1/24) * (a * r^4)
Simplifying this equation, we get:
r^3 = 24
Now, we need to check which value of r is consistent with both equations to find the correct common ratio.
For r = 3:
r^3 = 3^3 = 27 (which is not equal to 24)
For r = -3:
r^3 = (-3)^3 = -27 = -24 (which is equal to 24)
Therefore, the common ratio (r) of the GP is -3.
Now, we can find the fourth term (T4) by using the formula for the n-th term of a GP:
Tn = a * r^(n-1)
Since we want to find the fourth term, n = 4:
T4 = a * (-3)^(4-1)
T4 = a * (-3)^3
T4 = a * (-27)
T4 = -27a
Hence, the fourth term (T4) of the given GP is -27 times the first term (a).
This cannot be solved because:
In GP:
an = a1 ∙ r ⁿ ⁻ ¹
a2 = a1 ∙ r
a3 = a1 ∙ r²
a5 = a1 ∙ r⁴
a3 = 9 a1
a1 ∙ r² = 9 a1
Divide both sides by a1
r² = 9
r = ± √ 9
r = ± 3
a2 = a5 / 24
a1 ∙ r = a1 ∙ r⁴ / 24
Divide both sides by a1 ∙ r
1 = r³ / 24
Multiply both sides by 24
24 = r³
The result is:
24 = r³
24 = ( - 3 )³ = - 27
OR
24 = 3³ = 27
That is obviously not true.