Sequences using Surds

# Sequences using Surds

GCSE(H),

Sequences can be generated which involve the use of surds. Use the Surd Rules to generate terms in linear, quadratic or geometric surd sequences.

## Examples

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

Answer: 288sqrt(3)

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

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

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

Answer: 2sqrt(2)

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

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