A G.P has 6 terms. If the 3rd and 4th terms are 28 and -56 respectively,find: (a) the first term (b) the sum of the G.P
Is this a geometric progression? Exactly what does Government high school have to do with it?
To find the first term and the sum of the geometric progression (G.P.), we can use the formula for the nth term of a G.P.:
an = a * r^(n-1)
where:
an = nth term
a = first term
r = common ratio
n = number of terms
Given:
3rd term (a3) = 28
4th term (a4) = -56
Let's use this information to find the values of a and r:
Step 1: Relationship between a3 and a4
We know that a4 is the next term after a3, so we can use this relationship to find r:
a4 = a3 * r
-56 = 28 * r
Dividing both sides by 28:
r = -2
Step 2: Finding the first term (a)
To find the first term (a), we can substitute r = -2 and a3 = 28 into the formula for a3:
a3 = a * (-2)^(3-1)
28 = a * (-2)^2
28 = 4a
Dividing both sides by 4:
a = 7
So, the first term (a) is 7.
Step 3: Finding the sum of the G.P.
The sum of a G.P. can be calculated using the formula:
Sn = a * (1 - r^n) / (1 - r)
where:
Sn = sum of the first n terms
Here, n = 6 (since there are 6 terms in total).
Substituting the values of a = 7, r = -2, and n = 6 into the formula:
S6 = 7 * (1 - (-2)^6) / (1 - (-2))
Calculating the values inside the parentheses:
S6 = 7 * (1 - 64) / (1 + 2)
S6 = 7 * (-63) / 3
S6 = -147
So, the sum of the G.P. is -147.
To summarize:
(a) The first term (a) is 7.
(b) The sum of the G.P. is -147.
To find the first term and the sum of a geometric progression (G.P), we need to use the formulas for G.P.
In a G.P, each term is found by multiplying the preceding term by a constant ratio (r).
Let's start by finding the common ratio (r). Since the 3rd term is 28 and the 4th term is -56, we can use this information to find the common ratio.
We know that the 4th term (-56) is obtained by multiplying the 3rd term (28) by the common ratio (r):
-56 = 28 * r
To find the value of r, we can divide -56 by 28:
r = -56 / 28 = -2
Now that we have the common ratio (r), we can find the first term (a) and the sum of the G.P.
(a) Finding the first term:
To find the first term (a), we can use the formula:
a = (term)/(ratio)^(position - 1)
In this case, the 3rd term is given (28), and its position is 3. We substitute these values into the formula:
a = 28 / (-2)^(3-1)
a = 28 / (-2)^2
a = 28 / 4
a = 7
So, the first term (a) of the G.P is 7.
(b) Finding the sum of the G.P:
To find the sum of a G.P, we can use the formula:
Sum = a * (1 - r^n) / (1 - r)
In this case, we are given that there are 6 terms. So, n (the number of terms) is 6. Substituting this and the values we found above into the formula:
Sum = 7 * (1 - (-2)^6) / (1 - (-2))
Sum = 7 * (1 - 64) / (1 + 2)
Sum = 7 * (-63) / 3
Sum = -441 / 3
So, the sum of the G.P is -147.