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Find an explicit formula for the geometric sequence 120,60,30,15,dots Note: the first term should be a(1). a(n)=◻

Find an explicit formula for the geometric sequence.\newline120,60,30,15,…120, 60, 30, 15, \dots\newlineNote: the first term should be a(1)a(1).\newlinea(n)=□a(n)=\square

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Full solution

Q. Find an explicit formula for the geometric sequence.\newline120,60,30,15,…120, 60, 30, 15, \dots\newlineNote: the first term should be a(1)a(1).\newlinea(n)=□a(n)=\square
  1. Determine First Term and Ratio: To find an explicit formula for a geometric sequence, we need to determine the first term a(1)a(1) and the common ratio rr. The first term is given as 120120.
  2. Calculate Common Ratio: Next, we need to find the common ratio. The common ratio in a geometric sequence is found by dividing any term by the previous term. Let's divide the second term 6060 by the first term 120120.\newliner=60120=0.5r = \frac{60}{120} = 0.5
  3. Write Explicit Formula: Now that we have the first term a(1)=120a(1) = 120 and the common ratio r=0.5r = 0.5, we can write the explicit formula for the nnth term of the geometric sequence as:\newlinea(n)=a(1)×r(n−1)a(n) = a(1) \times r^{(n-1)}
  4. Substitute Values: Substituting the values of a(1)a(1) and rr into the formula, we get:\newlinea(n)=120×(0.5)(n−1)a(n) = 120 \times (0.5)^{(n-1)}\newlineThis is the explicit formula for the given geometric sequence.

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