he 12th term in a geometric sequence is 49 and the 14th therm is 2401, find the 15th term
Assume a(n) = C*b^n
a(12) = C*b^12 = 49
a(14) = C*b^14 = 2401
b^2 = 29 b = 7
C = 49/b^12 = 3.54*10^-9
a(15) = 2401*7 = _____
How do I round 2, 761 to the nearest hunderths
To find the 15th term in a geometric sequence, we can use the general formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Where:
an = the nth term
a1 = the first term
r = the common ratio
n = the term number
Given that the 12th term (a12) is 49 and the 14th term (a14) is 2401, we can set up the following equations:
a12 = a1 * r^(12-1) (Equation 1)
a14 = a1 * r^(14-1) (Equation 2)
We have two equations and two unknowns, a1 and r. Let's find a1 first.
From Equation 1:
49 = a1 * r^11
From Equation 2:
2401 = a1 * r^13
To eliminate a1, divide Equation 2 by Equation 1:
(2401/49) = (a1 * r^13) / (a1 * r^11)
Simplifying, we have:
(2401/49) = r^(13 - 11)
49 = r^2
Taking the square root of both sides, we get:
r = ±7
Now that we know the value of r, we can substitute it back into Equation 1 to find a1:
49 = a1 * 7^11
49 = a1 * 7^11
49 = 7^11 * a1
Dividing both sides by 7^11, we get:
a1 = 49 / 7^11
Now that we have a1 (the first term) and r (the common ratio), we can use the formula to find the 15th term, a15:
a15 = a1 * r^(15-1)
Substituting the values we found, we have:
a15 = (49 / 7^11) * 7^14
Simplifying, we get:
a15 = 49 * 7^(14 - 11)
a15 = 49 * 7^3
a15 = 49 * 343
a15 = 16,807
Therefore, the 15th term of the geometric sequence is 16,807.