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{[h(1)=-26],[h(n)=h(n-1)*(-9)]:} Find an explicit formula for h(n). h(n)=

{h(1)=−26,h(n)=h(n−1)⋅(−9)\begin{cases} h(1)=-26, h(n)=h(n-1)\cdot(-9) \end{cases}\newline Find an explicit formula for h(n)h(n).\newline\newline h(n)=h(n)=

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Q. {h(1)=−26,h(n)=h(n−1)⋅(−9)\begin{cases} h(1)=-26, h(n)=h(n-1)\cdot(-9) \end{cases}\newline Find an explicit formula for h(n)h(n).\newline\newline h(n)=h(n)=
  1. Given Information: We are given the first term of the sequence h(1)=−26h(1) = -26 and a recursive formula h(n)=h(n−1)×(−9)h(n) = h(n-1) \times (-9). To find an explicit formula for h(n)h(n), we need to recognize the pattern of the sequence.
  2. Recursive Formula: The sequence starts with h(1)=−26h(1) = -26. Using the recursive formula, we can find the next few terms to identify the pattern:\newlineh(2)=h(1)⋅(−9)=−26⋅(−9)=234h(2) = h(1) \cdot (-9) = -26 \cdot (-9) = 234\newlineh(3)=h(2)⋅(−9)=234⋅(−9)=−2106h(3) = h(2) \cdot (-9) = 234 \cdot (-9) = -2106\newlineWe can see that each term is the previous term multiplied by −9-9.
  3. Identifying the Pattern: Since the sequence is formed by multiplying the previous term by −9-9, this is a geometric sequence with the first term a1=−26a_1 = -26 and common ratio r=−9r = -9.
  4. Geometric Sequence: The explicit formula for a geometric sequence is given by:\newlinean=a1⋅r(n−1)a_n = a_1 \cdot r^{(n - 1)}\newlineSubstituting a1=−26a_1 = -26 and r=−9r = -9 into the formula gives us:\newlineh(n)=−26⋅(−9)(n−1)h(n) = -26 \cdot (-9)^{(n - 1)}

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