Algebra

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Special Sequences: Harder Geometric Sequences

You may find geometric sequences which have harder term-to-term rules, such as multiplying by a surd.

Some geometric sequences will require you to multiply by irrational numbers. To find the nth term in these sequences, we carry out the same process as with other sequences.

Let's look at an example:

$\sqrt{2}, \space 2,\space 2\sqrt{2},\space 4,\space 4\sqrt{2},\space 8...$

This is a geometric sequence, so each term is multiplied by the same number

But which number?

Each number is multiplied by $\sqrt{2}$

To produce the second term: $\sqrt{2} \times \sqrt{2}=2$ . Try multiplying each term by $\sqrt{2}$ to see if the pattern holds.

What would be the next term in the sequence?

If the next term is $8 \sqrt{2}$ , which term would come after this?

What is the next term in this sequence?

$2\sqrt{4}, 8, 8\sqrt{4}, 32, 32\sqrt{4}...$

Once we have identified the multiplier in the sequence, we can use this to find the nth term.

Let's pick up our example from earlier.

$\sqrt{2}, \space 2,\space 2\sqrt{2},\space 4,\space 4\sqrt{2},\space 8...$

We know that the multiplier here is $\sqrt{2}$

The next term in the sequence is found by multiplying the previous term by $\sqrt{2}$ . We can use this to find a phrase for the nth term of the sequence.

The nth term phrase begins with the first term

The first part of the nth term phrase is $\sqrt{2}$ , because this is the first term in the sequence. It will therefore come in the form $a_n=\sqrt{2} \times (\sqrt{2})^x$

For each new term, we multiply by $\sqrt{2}$

Therefore, each new term is produced by multiplying by a power of $\sqrt{2}$ . The third term is $\sqrt{2} \times\sqrt{2} \times \sqrt{2}=2\sqrt{2}$ . Similarly, the fourth term is $\sqrt{2} \times \sqrt{2}\times \sqrt{2}\times \sqrt{2}=4$ .

Therefore, each new term is a power of $\sqrt{2}$

The nth term can be expressed as $(\sqrt{2})^{n}$ .

If $a_n=(\sqrt{2})^n$ , what is the 10th term?

What is the expression for nth term in the following sequence?

$\sqrt{4}, 4, 4\sqrt{4}, 16, 16\sqrt{4}...$