the 2nd term of a progression is -7 and the 5th term is 56.find the common ratio and the sum of the first five terms
ar = -7
r^3 = 56/-7 = -8
Now you can easily fine a and r, and
S5 = a(r^5-1)/(r-1)
To find the common ratio, we can use the formula for the nth term of a geometric progression:
An = A1 * r^(n-1)
Given that the 2nd term A2 is -7 and the 5th term A5 is 56, we can set up two equations:
-7 = A1 * r^(2-1) [Equation 1]
56 = A1 * r^(5-1) [Equation 2]
We need to solve these equations simultaneously to find the values of A1 and r.
From Equation 1, we can rewrite it as:
A1 = -7 / r [Equation 3]
Substitute Equation 3 into Equation 2:
56 = (-7 / r) * r^4
56 = -7 * r^3
r^3 = -56/-7
r^3 = 8
Taking the cube root of both sides:
r = 2
Now that we have the value of the common ratio r as 2, we can substitute it into Equation 3:
A1 = -7 / 2
A1 = -3.5
The common ratio is 2 and the first term is -3.5.
To find the sum of the first five terms, we can use the formula for the sum of a geometric progression:
Sn = A1 * (1 - r^n) / (1 - r)
where Sn is the sum of the first n terms.
Substituting the values, we can find the sum of the first five terms:
S5 = -3.5 * (1 - 2^5) / (1 - 2)
S5 = -3.5 * (1 - 32) / -1
S5 = -3.5 * (-31) / -1
S5 = 108.5
Therefore, the common ratio is 2, and the sum of the first five terms is 108.5.
To find the common ratio and the sum of the first five terms, we need to determine the formula for the progression.
In a geometric progression, the formula for the nth term (Tn) is given by:
Tn = a * r^(n-1)
where:
- Tn refers to the nth term,
- a represents the first term, and
- r is the common ratio.
Using the given information, we can substitute the values for the second (T2) and the fifth (T5) terms into the formula:
T2 = a * r^(2-1)
= a * r
T5 = a * r^(5-1)
= a * r^4
Now, we can set up a system of equations to solve for the common ratio (r) and the first term (a).
Given: T2 = -7 and T5 = 56
Substituting the values:
-7 = a * r Eq. (1)
56 = a * r^4 Eq. (2)
To solve this system of equations, we can divide Eq. (2) by Eq. (1):
(56) / (-7) = (a * r^4) / (a * r)
-8 = r^3
So, r = -2.
Now, substituting r in Eq. (1):
-7 = a * (-2)
-7 = -2a
a = -7 / -2
a = 7/2 or 3.5
Thus, the common ratio (r) is -2, and the first term (a) is 3.5.
To find the sum of the first five terms, we can use the formula for the sum of the first 'n' terms of a geometric progression:
Sn = a * (1 - r^n) / (1 - r)
Substituting the values:
S5 = (3.5 * (1 - (-2)^5)) / (1 - (-2))
= (3.5 * (1 - 32)) / (1 + 2)
= (3.5 * (-31)) / 3
= -108.5
Therefore, the sum of the first five terms is -108.5.