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Woo-Jin and Kiran were asked to find an explicit formula for the sequence 64,16,4,164, 16, 4, 1, where the first term should be f(1)f(1)

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Q. Woo-Jin and Kiran were asked to find an explicit formula for the sequence 64,16,4,164, 16, 4, 1, where the first term should be f(1)f(1)
  1. Identify Geometric Pattern: We observe that each term in the sequence is the previous term divided by 44. This suggests that the sequence is geometric with the first term a=64a = 64 and the common ratio r=14r = \frac{1}{4}.
  2. Find Explicit Formula: To find the explicit formula for a geometric sequence, we use the formula f(n)=a⋅r(n−1)f(n) = a \cdot r^{(n-1)}, where aa is the first term, rr is the common ratio, and nn is the term number.
  3. Substitute Values: Substitute the values of aa and rr into the formula to get f(n)=64×(14)(n−1)f(n) = 64 \times (\frac{1}{4})^{(n-1)}.
  4. Simplify Formula: We can simplify the formula by noting that 6464 is 434^3, so we can write the formula as f(n)=43×(14)(n−1)f(n) = 4^3 \times (\frac{1}{4})^{(n-1)}.
  5. Apply Exponent Property: Using the property of exponents that (am)n=am∗n(a^m)^n = a^{m*n}, we can simplify the formula further to f(n)=43−n+1f(n) = 4^{3-n+1}.
  6. Final Explicit Formula: Simplify the exponent to get the final explicit formula: f(n)=44−nf(n) = 4^{4-n}.

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