what is the 7th geometric sequence when a10 is 9 and a13 is -72
9 9r 9r^2 9r^3
-72 = 9 r^3
r^3 = -72/9
now work back
a9 = 9/r
a8 = 9/r^2
a7 = 9/r^3
interesting
a7 = 9/r^3 = (9)(-9/72) = -81/72
or ,
using the standard a to be first term, r the common ratio
ar^12 = -72
ar^9 =9
divide them
r^3 = -8
r = -2 , then from ar^9 = 9
a = 9/(-2)^9 = -9/512
term(7) = ar^6
= (-9/512)(-2^6) = -576/512 = -9/8
To find the 7th term of the geometric sequence, we first need to determine the common ratio (r).
We are given two terms, a10 and a13. The general formula for the nth term of a geometric sequence is:
an = a1 * r^(n-1)
Using this formula, we can set up two equations using the given information:
a10 = a1 * r^(10-1) = a1 * r^9 = 9 ...........(equation 1)
a13 = a1 * r^(13-1) = a1 * r^12 = -72 ...........(equation 2)
To solve these equations, we will divide equation 2 by equation 1:
(a1 * r^12) / (a1 * r^9) = (-72) / 9
r^3 = -8
Taking the cube root of both sides, we get:
r = -2
Now that we have the common ratio (r = -2), we can find the first term (a1) by substituting this value back into equation 1:
a1 * (-2)^9 = 9
Simplifying the equation:
a1 * (-512) = 9
Dividing both sides by -512, we find:
a1 = 9 / (-512) = - 1/57
Finally, we can find the 7th term (a7) using the formula:
a7 = a1 * r^(7-1) = (-1/57) * (-2)^6
Simplifying, we get:
a7 = (-1/57) * 64
a7 = -64/57
Therefore, the 7th term of the geometric sequence is -64/57.