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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`


`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`


Substitute into the second term to find a


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)`

See also Calculating Exactly with Surds