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{[g(1)=-29],[g(n)=g(n-1)*(-4)]:} Find an explicit formula for g(n). g(n)=

{g(1)=−29,g(n)=g(n−1)⋅(−4)\begin{cases} g(1)=-29, g(n)=g(n-1)\cdot(-4) \end{cases} Find an explicit formula for g(n)g(n). g(n)=g(n)=

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Q. {g(1)=−29,g(n)=g(n−1)⋅(−4)\begin{cases} g(1)=-29, g(n)=g(n-1)\cdot(-4) \end{cases} Find an explicit formula for g(n)g(n). g(n)=g(n)=
  1. Given Information: We are given that g(1)=−29g(1) = -29 and that g(n)=g(n−1)×(−4)g(n) = g(n-1) \times (-4) for n > 1. This is a recursive definition of the function gg. To find an explicit formula, we need to determine how g(n)g(n) is related to the initial value g(1)g(1) when nn increases.
  2. Identifying Pattern: Let's look at the first few terms to see if we can identify a pattern. We know g(1)=−29g(1) = -29. Using the recursive formula:\newlineg(2)=g(1)×(−4)=−29×(−4)=116g(2) = g(1) \times (-4) = -29 \times (-4) = 116.\newlineg(3)=g(2)×(−4)=116×(−4)=−464g(3) = g(2) \times (-4) = 116 \times (-4) = -464.
  3. Geometric Sequence Analysis: We can see that each term is −4-4 times the previous term. This is a geometric sequence with the first term a=g(1)=−29a = g(1) = -29 and common ratio r=−4r = -4. The nnth term of a geometric sequence is given by the formula a⋅r(n−1)a \cdot r^{(n-1)}.
  4. Explicit Formula: Substituting the values of aa and rr into the formula for the nnth term, we get:\newlineg(n)=−29⋅(−4)(n−1)g(n) = -29 \cdot (-4)^{(n-1)}.\newlineThis is the explicit formula for g(n)g(n).

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