The graph shows the first three terms in a geometric sequence. (a.) what is the common ratio? explain it show how you determined your answer. (b.) what is the recursive formula for the sequence? (c.) what is iterative formula for the sequence? (d.) show how to use your answer from part (c) to find the eighth term of the sequence.

  (a) 2.5

  (b) a[1] = 2; a[n] = a[n-1]×2.5

  (c) a[n] = 2×2.5^(n-1)

  (d) a[8] = 1220.703125 (see below for the working)

(a) Sequential values read from the graph are 2, 5, 12.5. The common ratio is the ratio of a term to the one previous: 5/2 = 12/5 = 2.5

(b) A geometric sequence is defined as each term being the one before multiplied by the common ratio:

  a[n] = a[n-1]×2.5 . . . together with . . . a[1] = 2

(c) "Iteration" is usually describing a process in which the next value is obtained from the one before. Using this definition, the recursive formula and the iterative formula are exactly the same thing.

Usually, the "recursive formula" is contrasted to the "explicit formula" in which the n-th term is found directly. The explicit formula for a geometric sequence with first term a[1] and common ratio r can always be written ...

  a[n] = a[1]×r^(n-1)

For the sequence here, a[1] = 2 and r = 2.5, so this is ...

  a[n] = 2×2.5^(n-1)

We assume this question is asking for the explicit formula. If not, then the answer to part (c) is the same as the answer to part (b).

(d) Put 8 where n is in the formula and evaluate.

  a[8] = 2×2.5^(8-1) = 2×610.3515625 = 1220.703125

If you really mean to evaluate the recursive formula, then observe that ...

  a[8] = a[7]×2.5 = (a[6]×2.5)×2.5 = ((a[5]×2.5)×2.5)×2.5 = a[5]×2.5^3

     = (a[4]×2.5)×2.5^3 = ((a[3]×2.5)×2.5)×2.5^3 = (((a[2]×2.5)×2.5)×2.5)×2.5^3

     = a[2]×2.5^6 = a[1]×2.5×2.5^6 = 2×2.5^7

  a[8] = 1220.703125

Or, you can start with a[1] and work your way up to a[8].

  a[2] = 2×2.5 = 5

  a[3] = 5×2.5 = 12.5

  a[4] = 12.5×2.5 = 31.25

...

  a[7] = 195.3125×2.5 = 488.28125

  a[8] = 488.28125×2.5 = 1220.703125

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step-by-step explanation:

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