The third and sixth terms of a geometric progression(G.P) are 1/4 & 1/32 respectively. Find 1. The first term
2.the common ratio
3. The seventh term
What's not to understand? They tell you that
ar^2 = 1/4
ar^5 = 1/32
divide, and you have ar^5/ar^2 = r^3 = (1/32) / (1/4) = 1/8
so, r = 1/2
ar^2 = a(1/4) = 1/4, so a = 1
Now, the 7th term is the 6th term times the ratio (ar^6 = ar^5 * r)
= 1/32 * 1/2 = 1/64
I don't understand it
A Gp has 6 terms.if the 3rd and 4th terms are 28 and -56 respectively.find
a)the first term
b)the sum of Gp
T3=ar^2= 28
T4=ar^3= -56
r= -56/28; r=-2
From T3=ar^2 =28
T3=a(-2)^2= 28
4a=28
a=7
a=7, r=-2
The sum of a G.P
Sn =a(1-r^n)/1-r
Sn=7[(1-(-2)^n]/1-(-2)
Sn=7(1+2)^n/1+2
Sn=7(3)^n/3
Sn=21^n/3
Sn=7^n.
43
The 4th and 6th terms of a G.P are 1/10 and 1/160 respectively determine the common ratio and the first term
To find the first term, common ratio, and seventh term of a geometric progression, we can use the given information about the third term and sixth term.
In a geometric progression, each term is found by multiplying the previous term by a constant value called the common ratio (r).
1. Finding the common ratio (r):
Given that the third term is 1/4 and the sixth term is 1/32, we can set up the following equation:
3rd term = (1st term) * r^2
6th term = (1st term) * r^5
Substituting the given values:
1/4 = (1st term) * r^2 ---(equation 1)
1/32 = (1st term) * r^5 ---(equation 2)
Divide equation 2 by equation 1:
(1/32) / (1/4) = [(1st term) * r^5] / [(1st term) * r^2]
(1/32) * (4/1) = r^(5-2)
1/8 = r^3
Taking the cube root of both sides:
∛(1/8) = ∛(r^3)
1/2 = r
So, the common ratio (r) is 1/2.
2. Finding the first term (a):
Using equation 1, now that we have the value of r:
1/4 = (1st term) * (1/2)^2
1/4 = (1st term) * 1/4
1/4 = (1st term) * (1/4)
Therefore, the first term is 1.
3. Finding the seventh term:
Now that we know the first term (a = 1) and the common ratio (r = 1/2), we can find the seventh term using the formula for the nth term of a geometric progression: Tn = a * r^(n-1).
Plugging in the values:
T7 = 1 * (1/2)^(7-1)
T7 = 1 * (1/2)^6
T7 = 1 * (1/64)
T7 = 1/64
Therefore, the seventh term is 1/64.