2k+ 5k+1 + 10k+2 are three successive terms of a geometric sequence find the value(s) of k
I see 5 terms, not 3.
You will need brackets to identify the terms
Clearly, 10k+2 is 2(5k+1)
so, 5k+1 = 2(2k)
k = -1
The sequence is
-2 -4 -8 ...
To find the value(s) of k in the geometric sequence 2k, 5k+1, 10k+2, we need to determine the common ratio between consecutive terms.
In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio.
So, to find the common ratio, we divide the second term by the first term, and then divide the third term by the second term. The two ratios should be equal.
First, let's find the common ratio between the first and second terms:
Common Ratio (R1) = (5k+1)/(2k)
Next, let's find the common ratio between the second and third terms:
Common Ratio (R2) = (10k+2)/(5k+1)
Since the given sequence is a geometric sequence, these two ratios should be equal.
Setting up the equation:
(5k+1)/(2k) = (10k+2)/(5k+1)
Now, we can cross-multiply and solve for k.
(5k+1)(5k+1) = (10k+2)(2k)
(25k^2 + 10k + 1) = (20k^2 + 4k)
25k^2 + 10k + 1 = 20k^2 + 4k
25k^2 - 20k^2 + 10k - 4k = -1
5k^2 + 6k = -1
Rearranging the equation:
5k^2 + 6k + 1 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring the equation:
(5k + 1)(k + 1) = 0
Setting each factor to zero:
5k + 1 = 0 OR k + 1 = 0
Solving for k:
5k = -1 OR k = -1
k = -1/5 OR k = -1
Therefore, the possible values of k are k = -1/5 and k = -1.
To find the value(s) of k in the given geometric sequence, we can look at the ratio between consecutive terms.
The ratio between two consecutive terms in a geometric sequence is constant. Let's calculate the ratios between the given terms:
Ratio between the second and first term:
(5k+1) / (2k)
Ratio between the third and second term:
(10k+2) / (5k+1)
Since these ratios are constant, they should be equal. Therefore, we can set up an equation:
(5k+1) / (2k) = (10k+2) / (5k+1)
Let's solve this equation:
Cross-multiply to get rid of the denominators:
(5k+1)(5k+1) = (2k)(10k+2)
Expanding both sides:
25k^2 + 10k + 10k + 1 = 20k^2 + 4k
Combining like terms:
25k^2 + 20k + 1 = 20k^2 + 4k
Setting the equation to zero:
5k^2 - 16k + 1 = 0
Now, we can solve this quadratic equation using various methods - factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 5k^2 - 16k + 1 = 0, a = 5, b = -16, and c = 1. Substituting these values into the quadratic formula:
k = (-(-16) ± √((-16)^2 - 4(5)(1))) / (2(5))
Simplifying:
k = (16 ± √(256 - 20)) / 10
k = (16 ± √236) / 10
Finally, we can calculate the two possible values for k:
k = (16 + √236) / 10
k = (16 - √236) / 10
So, the two values of k in the given geometric sequence are:
k = (16 + √236) / 10
k = (16 - √236) / 10