How to find a specific term in a geometric sequence
Asked by admin @ in Mathematics viewed by 263 People
Given two terms from a geometric sequence, identify the first term and the common ratio: a5= 48 and a11= 3,072.
Answered by admin @
a5= 48 and a11= 3,072
Write out the formula for a geometric sequence: a_n = a_1*r^(n-1), with n
beginning at 1.
Then, using the given info:
a_5 = 48 is equivalent to a_48 = a_1*r^(5-1), or a_48 = a_1*r^4 = 48
a_11 = 3072 is equivalent to 3072 = a_1*r^10 = a_1*r^4*r^6 = 3072
Solving a_1*r^4 = 48 for a_1, we get 48 / r^4. Substitute this into the second equation to eliminate a_1:
a_1*r^4*r^6 = 3072 => (48 / r^4)*r^10 = 3072. Then 48*r^6 = 3072, and
r^6 = 3072 / 48 = 64. Thus, r = 6th root of 64 = 2
Now we must solve for a_1. Recall that 48 = a_1*r^4 and subst. 2 for r:
48 = a_1*2^4 => 48 = a_1*16. Then a_1 = 3.
The first term is 3 and the common ratio is 2.
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