The 2nd and 5th terms of a GP are -6 and 48, respectively.find the sum of the first four terms.
r^3 = 48/-6 = -8
r = -2
so a = 3
S4 = 3(16-1)/(-2-1) = -15
Why did the mathematician join the circus? Because he wanted to be an "addition-al" attraction!
Now, let's find that sum for you.
In a geometric progression (GP), each term is found by multiplying the previous term by a constant value, which is called the common ratio.
Let's call the common ratio "r".
We know that the 2nd term is -6 and the 5th term is 48.
So, based on the geometric progression formula, we can write:
-6 * r = 48
Now we can solve for "r":
r = 48 / -6
r = -8
Now that we have the common ratio, we can find the first term.
Using the formula for the nth term of a GP, we can write:
a₅ = a₁ * r^(n-1)
Plugging in the values we know:
48 = a₁ * (-8)^(5-1)
Simplifying it:
48 = a₁ * (-8)^4
48 = a₁ * 4096
a₁ = 48 / 4096
a₁ = 0.01172 (rounded to 5 decimal places)
Now that we have the first term, we can find the sum of the first four terms:
Sum = a₁ + a₂ + a₃ + a₄
Plugging in the values:
Sum = 0.01172 + (-6) + (-6) * (-8) + (-6) * (-8)^2
Simplifying it:
Sum = 0.01172 + (-6) + (-6) * (-8) + (-6) * 64
Sum = 0.01172 + (-6) + 48 + (-384)
Sum = -341.98828
So, the sum of the first four terms is approximately -341.98828. However, since we're dealing with a geometric progression, it would be more accurate to express the sum as -341.98828 (or -342) to acknowledge the decimal part.
To find the sum of the first four terms of a geometric progression (GP), we need to determine the common ratio (r) first.
Let's start by using the given information:
The second term (a₂) of the GP is -6.
The fifth term (a₅) of the GP is 48.
We can use the formula for the nth term of a GP: aₙ = a₁ * r^(n-1), where a₁ is the first term of the GP.
Using the second term, we can express it as a₂ = a₁ * r^(2-1) = a₁ * r.
Similarly, using the fifth term, we have a₅ = a₁ * r^(5-1) = a₁ * r^4.
Now, divide the two equations to eliminate a₁:
a₅ / a₂ = (a₁ * r^4) / (a₁ * r) => 48 / -6 = r^3.
Simplifying, we get:
-8 = r^3.
To find the value of r, we can take the cube root of both sides:
∛(-8) = ∛(r^3),
-2 = r.
Now that we have the common ratio (r = -2), we can find the first term (a₁) by substituting it into one of the equations. Let's use a₅ = a₁ * r^4:
48 = a₁ * (-2)^4,
48 = a₁ * 16,
a₁ = 48 / 16,
a₁ = 3.
So, the first term (a₁) of the GP is 3, and the common ratio (r) is -2.
Finally, we can calculate the sum of the first four terms (S₄) using the formula for the sum of a GP:
S₄ = a₁ * (1 - r^4) / (1 - r),
Substituting the values we found:
S₄ = 3 * (1 - (-2)^4) / (1 - (-2)),
S₄ = 3 * (1 - 16) / (1 + 2),
S₄ = 3 * (-15) / 3,
S₄ = -15.
Therefore, the sum of the first four terms of the geometric progression is -15.
To find the sum of the first four terms of a geometric progression (GP), we need to know the common ratio (r). With the given information, we can find the common ratio by using the formula:
\[r = \sqrt[n]{\frac{a_n}{a_1}}\]
Where \(a_1\) represents the first term, \(a_n\) represents the \(n^{th}\) term, and \(r\) represents the common ratio.
In this case, we are given the 2nd and 5th terms of the GP, which we can use to find the common ratio:
Given: \(a_2 = -6\) and \(a_5 = 48\)
Let's find the common ratio (r):
\[r = \sqrt[5-2]{\frac{48}{-6}}\]
Simplifying this expression:
\[r = \sqrt[3]{-8} = -2\]
Therefore, the common ratio (r) is -2.
Now, we can find the sum of the first four terms of the GP using the formula:
\[S_n = \frac{a_1(1 - r^n)}{1 - r}\]
Where \(S_n\) represents the sum of the first \(n\) terms, \(a_1\) represents the first term, \(r\) represents the common ratio, and \(n\) represents the number of terms.
Substituting the values we know into the formula:
\[S_4 = \frac{a_1(1 - (-2)^4)}{1 - (-2)}\]
\[S_4 = \frac{a_1(1 - 16)}{1 + 2}\]
\[S_4 = \frac{a_1(-15)}{3}\]
Since we don't know the value of \(a_1\), we cannot calculate the exact sum. However, we can express it in terms of \(a_1\) as:
\[S_4 = -5a_1\]
Therefore, the sum of the first four terms of the GP is \(S_4 = -5a_1\).