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{[f(1)=-71],[f(n)=f(n-1)*4.2]:} Find an explicit formula for f(n). f(n)=

{f(1)=−71,f(n)=f(n−1)×4.2\begin{cases} f(1)=-71, f(n)=f(n-1)\times 4.2 \end{cases} Find an explicit formula for f(n)f(n).f(n)=f(n)=

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Q. {f(1)=−71,f(n)=f(n−1)×4.2\begin{cases} f(1)=-71, f(n)=f(n-1)\times 4.2 \end{cases} Find an explicit formula for f(n)f(n).f(n)=f(n)=
  1. Given information: We are given that f(1)=−71f(1) = -71 and that f(n)f(n) is 4.24.2 times f(n−1)f(n-1). To find an explicit formula for f(n)f(n), we need to express f(n)f(n) in terms of nn without the recursion.
  2. Identifying sequence type: Since f(n)=f(n−1)×4.2f(n) = f(n-1) \times 4.2, we can see that each term is 4.24.2 times the previous term. This is a geometric sequence with the first term f(1)=−71f(1) = -71 and the common ratio r=4.2r = 4.2.
  3. Geometric sequence formula: The explicit formula for the nnth term of a geometric sequence is given by f(n)=a⋅r(n−1)f(n) = a \cdot r^{(n-1)}, where aa is the first term and rr is the common ratio.
  4. Substitute values: Substitute a=−71a = -71 and r=4.2r = 4.2 into the formula to get f(n)=−71×4.2(n−1)f(n) = -71 \times 4.2^{(n-1)}.

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