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Bernardo and Ogechi were asked to find an explicit formula for the sequence 1,8,64,5121, 8, 64, 512 where the first term should be h(1)h(1)

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Q. Bernardo and Ogechi were asked to find an explicit formula for the sequence 1,8,64,5121, 8, 64, 512 where the first term should be h(1)h(1)
  1. Observe Sequence Pattern: We need to observe the pattern of the sequence to find the explicit formula. The sequence is 1,8,64,5121, 8, 64, 512. Let's look at the ratio of each term to its previous term to determine if it's a geometric sequence.
  2. Calculate Ratios: Calculate the ratio between the second term and the first term: 8/1=88 / 1 = 8. Calculate the ratio between the third term and the second term: 64/8=864 / 8 = 8. Calculate the ratio between the fourth term and the third term: 512/64=8512 / 64 = 8. Since the ratio is constant, it is a geometric sequence with a common ratio of 88.
  3. General Form of nth Term: The general form of the nnth term for a geometric sequence is given by 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. For this sequence, a=1a = 1 and r=8r = 8.
  4. Substitute Values: Substitute the values of aa and rr into the formula to get the explicit formula for the sequence: h(n)=1×8(n−1)h(n) = 1 \times 8^{(n-1)}.
  5. Simplify Formula: Simplify the formula to get the final explicit formula: h(n)=8(n−1)h(n) = 8^{(n-1)}.

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