4m + 1, m + 1 and 10 - m are the first three terms of a geometric sequence. Find t4.
to be an GS
(m+1)/(4m+1) = (10-m)/(m+1)
cross-multiply
m^2 + 2m + 1 = 10 +39m - 4m^2
5m^2 - 37m -9 = 0
m = (37 ± √1549)/10
you don't state whether you want "exact" values or if you want the fourth term in terms of m
you could simply say ...
since the common ratio is (m+1)/(4m+1)
term(4) = r (term(3) )
= (10-m)(m+1)/(4m+1)
To find the fourth term, t4, of the geometric sequence, we need to determine the common ratio, r.
We know that the first three terms of the geometric sequence are 4m + 1, m + 1, and 10 - m.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.
So, we can set up the following equations to find the common ratio:
(m + 1) / (4m + 1) = (10 - m) / (m + 1)
Cross-multiplying, we get:
(m + 1)^2 = (4m + 1)(10 - m)
Expanding and simplifying the equation gives:
m^2 + 2m + 1 = 40m - 4m^2 - 10 + m
Rearranging and combining like terms, we have:
5m^2 + 37m - 11 = 0
To solve this quadratic equation, we can use the quadratic formula:
m = [ -b ± sqrt(b^2 - 4ac) ] / 2a
Plugging in the values a = 5, b = 37, and c = -11, we get:
m = [ -37 ± sqrt(37^2 - 4*5*(-11)) ] / (2*5)
Simplifying further, we have:
m = [ -37 ± sqrt(1369 + 220) ] / 10
m = [ -37 ± sqrt(1589) ] / 10
Using a calculator, we find two values for m:
m ≈ 0.213 or m ≈ -7.813
Now that we have the potential values for m, we can find the common ratio, r.
Using the equation (m + 1) / (4m + 1) = (10 - m) / (m + 1), we can substitute one of the values of m and solve for r.
Taking the first value of m (m ≈ 0.213):
(0.213 + 1) / (4*0.213 + 1) = (10 - 0.213) / (0.213 + 1)
Simplifying, we get:
1.213 / 1.852 = 9.787 / 1.213
The common ratio, r, is found to be approximately 9.787 / 1.213 ≈ 8.07.
Now, we can find the fourth term, t4, by multiplying the third term by the common ratio:
t4 = (10 - m) * r
Substituting in the first value of m (m ≈ 0.213) and the common ratio (r ≈ 8.07), we get:
t4 ≈ (10 - 0.213) * 8.07
Calculating this expression yields:
t4 ≈ 81.658
Therefore, the fourth term, t4, is approximately 81.658.