(1) The 6th term of a G.P is -7 over 32. find its common ratio if its first term is 28.
term n = a r^(n-1)
term 6 = a r^5 = -7/32 and a = 28
-7 / 32 = 28 r^5 = -7/2^5
r^5 =(-7/28)(1/32) = (-1/4)(1/2)^5
r= (1/2)(-1/4)^0.2
I get -.38
??? 7/32 = 0.21875
28 * .38^5 = 0.22185
what went wrong?
r^5 = (-7/28)(1/32) = -(1/4)(1/2)^5 = -1/2^7
r = -1/2^1.4 = -0.37893
check: 28*0.37893^5 = 0.21875
To find the common ratio of a geometric progression (G.P.), we can use the formula:
a_n = a * r^(n-1)
Where:
a_n is the nth term of the G.P.
a is the first term of the G.P.
r is the common ratio of the G.P.
n is the position of the term we are interested in.
In this case, we are given:
a_n = -7/32
a = 28
n = 6
Plugging these values into the formula, we can write the following equation:
-7/32 = 28 * r^(6-1)
Simplifying the equation, we have:
-7/32 = 28 * r^5
To solve for r, we can isolate it by dividing both sides of the equation by 28:
(-7/32) / 28 = r^5
-7/896 = r^5
Now, we can take the fifth root of both sides to solve for r:
r = ( -7/896 )^(1/5)
Using a calculator or computer, you can evaluate this expression to find the value of r.