If 7 and 189 are the first and fourth terms of a geometric respectively, find the sum of the first three terms of the progression?
7 r^3 = 189 ... r^3 = 27 ... r = 3
7 + 21 + 63 = ?
91
To find the sum of the first three terms of a geometric progression, we first need to determine the common ratio (r) of the progression.
Given that 7 is the first term (a₁) and 189 is the fourth term (a₄), we can use the formula for the nth term of a geometric progression to find the common ratio (r).
The formula for the nth term of a geometric progression is given by:
aₙ = a₁ * r^(n-1)
Plugging in the values:
a₄ = 7 * r^(4-1) = 7 * r^3
We can rearrange this equation to find the common ratio (r):
r = (a₄ / a₁)^(1/3)
Substituting the values of a₄ = 189 and a₁ = 7 into the equation:
r = (189 / 7)^(1/3) = 3
Therefore, the common ratio (r) of the geometric progression is 3.
To find the sum of the first three terms (S₃) of the progression, we use the formula:
S₃ = a₁ + a₂ + a₃
Given that a₁ = 7 and r = 3, we can substitute these values into the formula:
S₃ = 7 + 7 * 3 + 7 * 3^2
= 7 + 21 + 63
= 91
Therefore, the sum of the first three terms of the geometric progression is 91.
To find the sum of the first three terms of a geometric progression, we can use the formula:
S = a * (1 - r^n) / (1 - r),
where S is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, we are given the first term, a = 7, and the fourth term, t₄ = 189. Since the fourth term is given, we can use it to find the common ratio, r.
We can write the formula for the fourth term:
tₙ = a * r^(n-1).
Substituting n = 4 and t₄ = 189, we get:
189 = 7 * r^(4-1),
189 = 7 * r^3.
Dividing both sides of the equation by 7:
27 = r^3.
Taking the cube root of both sides:
r = ∛(27),
r = 3.
Now, we have the common ratio, r = 3. We can use this to find the sum of the first three terms, S₃.
Using the formula, we have:
S₃ = a * (1 - r^3) / (1 - r).
Substituting a = 7 and r = 3:
S₃ = 7 * (1 - 3^3) / (1 - 3).
Performing the calculations:
S₃ = 7 * (1 - 27) / (1 - 3),
S₃ = 7 * (-26) / (-2),
S₃ = 7 * 13,
S₃ = 91.
Therefore, the sum of the first three terms of the geometric progression is 91.