The first term of the given sequence is a(1) = 128
The formula for the nth term of a geometric sequence is:
a(n) = a(1) . rⁿ⁻¹
In the above formula, a(1) refers to the first term while r refers to the common ratio.
Information from the problem.
a(9) = 1/2 and a(12) = 1/16
Substitute n = 9 into the formula, we get:
a(9) = a(1) . r⁸
1/2 = a(1) . r⁸
Substitute n = 12 into the formula, we get:
a(12) = a(1) . r¹¹
1/16 = a(1) . r¹¹
Divide a(12) by a(9):
1/16 : 1/2 = ( a(1) . r¹¹ ) : (a(1) . r⁸ )
Since a(1) terms are cancelled out, we have:
2 : 16 = r¹¹ : r⁸
1/8 = r³
r = 1/2
Substitute r = 1/2 into a(9) :
a(9) = a(1) . r⁸
1/2 = a(1) . (1/2)⁸
a(1) = 1/2 : (1/2)⁸ = (1/2)¹⁻⁸
= (1/2)⁻⁷ = 2⁷
= 128
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