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d(n)=-5((1)/(2))^(n-1) What is the 3^("rd ") term in the sequence?

d(n)=−5(12)n−1 d(n)=-5\left(\frac{1}{2}\right)^{n-1} \newlineWhat is the 3rd  3^{\text {rd }} term in the sequence?

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Full solution

Q. d(n)=−5(12)n−1 d(n)=-5\left(\frac{1}{2}\right)^{n-1} \newlineWhat is the 3rd  3^{\text {rd }} term in the sequence?
  1. Understanding the sequence formula: Understand the given sequence formula.\newlineThe sequence is defined by the formula d(n)=−5(12)n−1d(n)=-5\left(\frac{1}{2}\right)^{n-1}, which suggests that it is a geometric sequence with a common ratio of 12\frac{1}{2} and a starting term multiplied by −5-5.
  2. Finding the 33rd term: Plug in the value of nn to find the 33rd term.\newlineTo find the 33rd term, we substitute n=3n=3 into the formula.\newlined(3)=−5(12)3−1d(3) = -5\left(\frac{1}{2}\right)^{3-1}
  3. Calculating the exponent: Calculate the exponent part of the formula. (12)(3−1)=(12)2(\frac{1}{2})^{(3-1)} = (\frac{1}{2})^2
  4. Simplifying the exponent: Simplify the exponent.\newline(12)2=14(\frac{1}{2})^2 = \frac{1}{4}
  5. Multiplying by −5-5 to get the 33rd term: Multiply the result of the exponent by −5-5 to get the 33rd term.\newlined(3)=−5×(14)d(3) = -5 \times \left(\frac{1}{4}\right)
  6. Performing the multiplication: Perform the multiplication to find the 3rd3^{\text{rd}} term.d(3)=−54d(3) = -\frac{5}{4}

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