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The pattern is adding numbers together that are being multiplied by 4 each time. This tells me that you’re taking 4 to the nth power, and adding each product from 0-n.
(I started with 9 as the first power and worked down, but it works the other way as well) the reason I started at 4^0 and went to 4^9 is because the first number in the series is 1, which tells me 4^0 and it wants to S10, which tells me that 4^9 has to be the last power added to make it S10. Please comment if you want me to try to clarify for you more.
Ver imagen broffutt

The sequence for [tex]1,4,16,64,...[/tex] will be [tex]2,62,144[/tex] .

What is geometric sequence  ?

Geometric sequence a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

[tex]a+ar+ar^2+ar^3 ...[/tex]

Here, [tex]a=[/tex] First term

         [tex]r=[/tex] common ratio

Next term formula, [tex]a_{n} =a_{1}*r^{n-1}[/tex]

Here, [tex]a_{1} =[/tex] First term

        [tex]a_{n} =[/tex] Term to be find

We have,

[tex]1,4,16,64,...[/tex] which is Geometric sequence.

First find common ratio;

So,

[tex]a_{2} =a_{1}*r^{2-1}[/tex]

[tex]4 =1*r[/tex]

⇒ [tex]r=4[/tex]

Now, Using  [tex]a_{n} =a_{1}*r^{n-1}[/tex] we will find out [tex]10th[/tex] term i.e. [tex]n=10[/tex],

[tex]a_{10} =a_{1}*r^{10-1}[/tex]

[tex]a_{10} =1*4^{9}[/tex]

[tex]a_{10} =2,62,144[/tex]

Hence, we can say that the sequence for [tex]1,4,16,64,...[/tex] is [tex]2,62,144[/tex] .. which is find out using Geometric sequence.

To know more about Geometric sequence click here

https://brainly.com/question/11266123

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