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)`