Question

Write an explicit and a cursive equation for the geometric sequence

a. 2, 10, 50, 250, 1250, . . .

Answer 1

• explicit: a(n) = 2·5^(n-1)
• recursive: a(1) = 2; a(n) = a(n-1)×5

Explanation:

The given sequence is exponential with a first term of a1 = 2 and a common ratio of r = 10/2 = 5.

The explicit equation for an exponential sequence is ...

a(n) = a(1)×r^(n -1)

So, for the given parameters, the explicit equation is ...

a(n) = 2×5^(n -1)

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The recursive equation for any series defines the next term as a function of previous terms. For a geometric sequence the next term is the previous term multiplied by the common ratio. For this sequence, the recursive definition is ...

a(1) = 2

a(n) = 5×a(n-1)

Answer 2

Explanation:

Explicit

a_n = a1 * r^(n - 1)

Find r by dividing term (n)/(n - 1)

r = 250 / 50

r = 5

a_n = a * r^(n - 1)

Recursive

a_n = a_(n-1)*r

Try an example

Find a_6

a_6 = 2 * 5^(6 - 1)

a_6 = 2 * 5^5

a_6 = 2 * 3125

a_6 = 6250

recursive

a_n = a_(n - 1)*r

r = 5

n = 6

a_n = 1250 * 5

a_n = 6250

The explicit method looks a whole lot easier, but not for a machine made by Dell and programed by Microsoft. A computer doesn't really mind doing a whole lot of calculations that are repetitive. And in many cases recursive is easier to program and is faster.