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

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)`

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)`

See also Calculating Exactly with Surds

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