<h3>
Answer:</h3>
(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)
<h3>
Step-by-step explanation:</h3>
(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 <em>explicit formula</em>. 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
