the 5th term in a geometric sequence is 160. The 7th term is 40. What are the possible values of the 6th term of the sequence
+70
70
+80
80
a 5 = 160
a 6 = a 5 * q = 160 * q
a 7 = a 6 * q = a 5 * q * q = 160 * q ^ 2 = 40
160 q ^ 2 = 40 Divide both sides by 40
4 q ^ 2 = 1 Divide both sides by 4
q ^ 2 = 1 / 4 Take square root of both sides
q = + OR - 1 / 2
All number in progression is positive so q = 1 / 2
a 6 = a 5 * q = 160 * q
a 6 = 160 * 1 / 2 = 80
80
What is the answer???
80 or +-80??
Well, I'm not much of a mathematician, but I'll try to give you an answer with a dash of humor!
Let's call the first term 'a', and the common ratio 'r'. Since it's a geometric sequence, we can use the formula for the nth term: aₙ = a₁ * r^(n-1).
From the information given, we know that a₅ = 160. So, plugging in the values: 160 = a * r^(5-1).
And we also know that a₇ = 40. Using the formula again, 40 = a * r^(7-1).
Now comes the fun part - solving these equations simultaneously! By dividing the second equation by the first equation, we can eliminate 'a':
(40/160) = (a_ * r^(7-1)) / (a * r^(5-1))
Now simplify that and you'll get 1/4 = r².
So, r = √(1/4), which means r can be either +1/2 or -1/2.
Now we can substitute these values into one of the original equations to find the possible values for 'a'.
If we use a₅ = 160 -> 160 = a * (+1/2)^(5-1) or 160 = a * (-1/2)^(5-1).
And solving these equations will give you the possible values of 'a'! So, the answer is a mixture of math and logic.
To find the possible values of the 6th term in a geometric sequence, we need to determine the common ratio (r) of the sequence.
1. We are given that the 5th term is 160, so we can write the equation: a₅ = a₁ * r⁴ = 160.
2. Similarly, the 7th term is 40, so we can write the equation: a₇ = a₁ * r⁶ = 40.
To find r, we can divide these two equations:
a₇ / a₅ = (a₁ * r⁶) / (a₁ * r⁴)
40 / 160 = r²
1 / 4 = r²
Taking the square root of both sides, we find two possible values for r: r = +/- (1/2).
Now, we can find the 6th term (a₆) by multiplying the 5th term by the common ratio:
a₆ = a₅ * r = 160 * (1/2) = 80
So, the possible values of the 6th term are +80 and -80. Therefore, the correct answer is +80.