Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are 40 and 5000, respectively.

OPTIONS ---
an = 8 • 5n - 2
an = 8 • 5n + 3
an = 8 • 5n - 1
an = 8 • 5n + 1

Answer 1

Given:
2nd term : 40
5th term : 5,000

The correct format of the given choices are:
an = 8 • 5^(n - 2)
an = 8 • 5^(n + 3)
an = 8 • 5^(n - 1)
an = 8 • 5^(n + 1)

They are the exponents. I individually substituted n by 2 and 5 to get the correct value of the corresponding term and the correct explicit rule is:

a(n) = 8 * 5^(n-1)

a(2) = 8 * 5^(2-1) = 8 * 5^1 = 8 * 5 = 40
a(5) = 8 * 5^(5-1) = 8 * 5^4 = 8 * 625 = 5,000

Answer 2

`a_n=8.5^(n-1)`

Explanation:

Since, the explicit rule of geometric sequence is,

`a_n=a.r^(n-1)`

Where, a is the first term,

r is the common ratio,

n is the number of term,

Given,

The second term is 40,

`\implies ar^(2-1)=40`

`ar = 40-----(1)`

Also, the fifth term is 5000,

`\implies ar^(5-1)=5000`

`ar^4 = 5000`

`(ar)r^3=5000`

From equation (1),

`40r^3 = 5000`

`r^3=125`

`\implies r = 5`

Again from equation (1),

We get,

a = 8

Hence, the explicit rule for the give geometric sequence is,

`a_n=8.5^(n-1)`