akshatgupta87 wrote:Q) A sequence of numbers (geometric sequence) is given by the expression: g(n)=5*(-1/2)^n . If the sequence begins with n = 1, what are the first two terms for which |gn - g(n+1)| < 1/1000?
A) g10, g11
B) g11, g12
C) g12, g13
D) g13, g14
E) g14, g15
OA is D
Someone please explain.
~Thanks,
Akshat
We can plug in the answers.
It's a good idea to know the powers of 2 up to 2¹�.
2¹� = 1024.
2¹¹ ≈ 2000.
2¹² ≈ 4000.
2¹³ ≈ 8000.
2¹� ≈ 16,000.
Answer choice C: g�₂, g�₃
|g�₃ - g�₂| < 1/1000
|5*(-1/2)¹³ - 5*(-1/2)¹²| < 1/1000
5 * |-1/8000 - 1/4000| < 1/1000
5 * (3/8000) < 1/1000
15/8 < 1.
Doesn't work.
For the left side to DECREASE, the exponents must INCREASE.
Answer choice D: g�₄, g�₃
|g�₄ - g�₃| < 1/1000
|5*(-1/2)¹� - 5*(-1/2)¹³| < 1/1000
5 * |1/16,000 + 1/8000 | < 1/1000
5 * (3/16,000) < 1/1000
15/16 < 1.
Success!
The correct answer is
D.
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