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{[h(1)=-17],[h(n)=h(n-1)*0.2]:} Find an explicit formula for h(n). h(n)=

{h(1)=−17,h(n)=h(n−1)×0.2\begin{cases} h(1)=-17, h(n)=h(n-1)\times 0.2 \end{cases} Find an explicit formula for h(n)h(n). h(n)=h(n)=

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Q. {h(1)=−17,h(n)=h(n−1)×0.2\begin{cases} h(1)=-17, h(n)=h(n-1)\times 0.2 \end{cases} Find an explicit formula for h(n)h(n). h(n)=h(n)=
  1. Given Information: We are given the first term of the sequence h(1)=−17h(1) = -17 and a recursive formula h(n)=h(n−1)×0.2h(n) = h(n-1) \times 0.2. To find an explicit formula, we need to recognize the type of sequence we are dealing with.
  2. Identifying the Sequence Type: Since each term is obtained by multiplying the previous term by a constant factor 0.20.2), this sequence is geometric.
  3. Geometric Sequence Formula: For a geometric sequence, the nth term is given by the formula h(n)=a⋅r(n−1)h(n) = a \cdot r^{(n-1)}, where aa is the first term and rr is the common ratio.
  4. Substituting Values: We already know the first term a=h(1)=−17a = h(1) = -17 and the common ratio r=0.2r = 0.2. Now we can substitute these values into the formula for the nnth term of a geometric sequence.
  5. Final Explicit Formula: Substituting the values, we get h(n)=−17⋅(0.2)n−1h(n) = -17 \cdot (0.2)^{n-1}.

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