Sequences using Surds

## Sequences using Surds

Sequences can be generated which involve the use of surds. Surds can be used in generating terms in linear, quadratic or geometric sequences.

## Example 1

What is the next number in this geometric sequence? 2sqrt(3), 12, 24sqrt(3), 144

The first term is 2sqrt(3)=ar^1

Rearrange to a=frac(2sqrt(3))(r)

Substituting into the second term (12=ar^2): 12 = frac(2sqrt(3))(r) xxr^2

12=2sqrt(3)r

r = frac(12)(2sqrt(3)) = 2sqrt(3)

Substitute r into the first term to get a = 1

Check using, say, the fourth term:

144 = (1)(2sqrt(3))^4

Generate the fifth term U_5 = (1)(2sqrt(3))^5 = 288sqrt(3)

Answer: 288sqrt(3)

## Example 2

The second and fourth terms of a geometric sequence are 2 and 1. What is the first term?

The second term is 2 = ar^2, rearrange to a=frac(2)(r^2)

Substitute into the fourth term: 1=ar^4

1 = (frac(2)(r^2))r^4=2r^2

r=frac(1)(sqrt(2))

Substitute into the second term to find a

2=a(frac(1)(sqrt(2)))^2=frac(a)(2)

Therefore a=4

Check the value of a and r for the second and fourth terms:

For the 2nd term: 2=(4)(frac(1)(sqrt(2)))^2

For the 4th term: 1=(4)(frac(1)(sqrt(2)))^4

First term is given by u_1=frac(4)(sqrt(2))=frac(4(sqrt(2)))(2)=2sqrt(2)

Answer: 2sqrt(2)